Physics Review Notes - Tom · PDF filePhysics Review Notes 2007 ... 3 Chapter 3 — Projectile Motion ... Chapter 4 — Newton

Physics Review Notes2007–2008

Tom StrongScience Department

Mt Lebanon High [email protected]

June, 2008

The most recent version of this can be found at http://www.tomstrong.org/physics/

Chapter 1 — About Science . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1Chapter 2 — Linear Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3Chapter 3 — Projectile Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5Chapter 4 — Newton’s First Law of Motion - Inertia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6Chapter 5 — Newton’s Second Law of Motion — Force and Acceleration . . . . . . . . . . . . . . . . . . . . . 7Chapter 6 — Newton’s Third Law of Motion - Action and Reaction . . . . . . . . . . . . . . . . . . . . . . . . 8Chapter 7 — Momentum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9Chapter 8 — Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10Chapter 9 — Circular Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12Chapter 10 — Center of Gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13Chapter 11 — Rotational Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14Chapter 12 — Universal Gravitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16Chapter 13 — Gravitational Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17Chapter 14 — Satellite Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18Chapter 25 — Vibrations and Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19Chapter 26 — Sound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21Chapter 27 — Light . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23Chapter 28 — Color . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25Chapter 29 — Reflection and Refraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27Chapter 30 — Lenses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29Chapter 31 — Diffraction and Interference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31Chapter 32 — Electrostatics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33Chapter 33 — Electric Fields and Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34Chapter 34 — Electric Current . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35Chapter 35 — Electric Circuits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37Chapter 36 — Magnetism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38Chapter 37 — Electromagnetic Induction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40Variables and Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42Mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45Data Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51Final Exam Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

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These notes are meant to be a summary of important points covered in the Physics class at Mt. Lebanon High School.They are not meant to be a replacement for your own notes that you take in class, nor are they a replacement for yourtextbook.

This is a work in progress and will be changing and expanding over time. I have attempted to verify the correctnessof the information presented here, but the final responsibility there is yours. Before relying on the information in thesenotes please verify it against other sources.

Physics 2007–2008 Mr. Strong

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Chapter 1 — About Science

1.1 The Basic Science — PhysicsPhysics is the most basic of the living and non-living sciences. All other sciences are built on a knowledge of physics. Wecan understand science in general much better if we understand physics first.

1.2 Mathematics — The Language of SciencePhysics equations are systems of connections following all of the rules of logic. They not only tell us what is connected towhat, they tell us what we can ignore. Mathematical equations are unambiguous, they don’t have the multiple meaningsthat often confuse the discussions of ideas expressed in common language.

1.3 The Scientific MethodThe scientific method is a process that is extremely effective in gaining, organizing, and applying new knowledge. Thegeneral form of the scientific method is:

1. Recognize a problem through observation.

2. Make an educated guess (a hypothesis) about the answer.

3. Predict the consequences of the hypothesis.

4. Perform one or more experiments to test the hypothesis.

5. Analyze and interpret the results of the experiment(s)

6. Share your conclusions with others in a way that they can independently verify your results.

There are many ways of stating the scientific method, this is just one of them. Some steps above are sometimes combined,at other times one step may be divided into two or more. The important part is not the number of steps listed or theirgrouping but is instead the general process and the emphasis on deliberate reproduceable action.

1.4 The Scientific AttitudeIn science a fact is just a close agreement by competent observers who make a series of observations of the samephenomenon. In other words, it is what is generally believed by the scientific community to be true.

A hypothesis is an educated guess that is only presumed to be factual until verified or contradicted through exper-iment.

When hypotheses are tested repeatedly and not contradicted they may be accepted as fact and are then known aslaws or principles.

If a scientist finds evidence that contradicts a law, hypothesis, or principle then (unless the contradicting evidenceturns out to be wrong) that law, hypothesis, or principle must be changed to fit the new data or abandoned if it can notbe changed.

In everyday speech a theory is much like a hypothesis in that it generally indicates something that has yet to beverified. In science a theory is instead the result of well-tested and verified hypotheses about the reasons for certainobserved behaviors. Theories are refined as new information is obtained.

Scientific facts are statements that describe what happens in the world that can be revised when new evidence isfound, scientific theories are interpretations of the facts that explain the reasons for what happens.

1.5 Scientific Hypotheses Must Be TestableWhen a hypothesis is created it is more important that there be a means of proving it wrong than there be a means ofproving it correct. If there is no test that could disprove it then a hypothesis is not scientific.

1.6 Science, Technology, and SocietyScience is a method of answering theoretical questions, technology is a method for solving practical problems. Scientistspursue problems from their own interest or to advance the general body of knowledge, technologists attempt to design,create or build something for the use or enjoyment of people.


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1.7 Science, Art, and ReligionScience, art, and religion all involve the search for order and meaning in the world, and while they each go back thousandsof years and overlap in several ways, all three exist for different purposes. Art works to describe the human experienceand communicate emotions, science describes natural order and predicts what is possible in nature, and religion involvesnature’s purpose.

Accuracy vs. Precision• Accuracy describes how close a measured value is to the true value of the quantity being measured

Problems with accuracy are due to error. To avoid error:

– Take repeated measurements to be certain that they are consistent (avoid human error)– Take each measurement in the same way (avoid method error)– Be sure to use measuring equipment in good working order (avoid instrument error)

• Precision refers to the degree of exactness with which a measurement is made and stated.

– 1.325 m is more precise than 1.3 m– lack of precision is usually a result of the limitations of the measuring instrument, not human error or lack of

calibration– You can estimate where divisions would fall between the marked divisions to increase the precision of the

measurement

Counting Significant Figures in a NumberRule ExampleAll counted numbers have an infinite numberof significant figures

10 items, 3 measurements

All mathematical constants have an infinitenumber of significant figures

1/2, π, e

All nonzero digits are significant 42 has two significant figures; 5.236 has fourAlways count zeros between nonzero digits 20.08 has four significant figures; 0.00100409 has sixNever count leading zeros 042 and 0.042 both have two significant figuresOnly count trailing zeros if the number con-tains a decimal point

4200 and 420000 both have two significant figures; 420.has three; 420.00 has five

For numbers in scientific notation apply theabove rules to the mantissa (ignore the expo-nent)

4.2010× 1028 has five significant figures

Counting Significant Figures in a CalculationRule ExampleWhen adding or subtracting numbers, find the number which is knownto the fewest decimal places, then round the result to that decimalplace.

21.398 + 405 − 2.9 = 423 (3significant figures, roundedto the ones position)

When multiplying or dividing numbers, find the number with the fewestsignificant figures, then round the result to that many significant fig-ures.

0.049623 × 32.0/478.8 =0.00332 (3 significant figures)

When raising a number to some power count the number’s significantfigures, then round the result to that many significant figures.

5.82 = 34 (2 significant fig-ures)

Mathematical constants do not influence the precision of any compu-tation.

2× π × 4.00 = 25.1 (3 signif-icant figures)

In order to avoid introducing errors during multi-step calculations, keepextra significant figures for intermediate results then round properlywhen you reach the final result.


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Chapter 2 — Linear Motion

2.1 Motion is Relative

To measure the motion of an object it muse be measured relative to something else, in other words some reference needsto be there to measure the motion against. You can measure the motion of a car moving down the highway, but if youmeasure it from another moving car you will get very different measurements than if you are a person standing besidethe road.

2.2 Speed

A moving object travels a certain distance in a given time. Speed is a measure of how fast something is moving or howfast distance is covered. It is measured in terms of length covered per unit time so you will encounter units like metersper second or miles per hour when working with speed.

Instantaneous SpeedThe speed that an object is moving at some instant is the instantaneous speed of that object. If you are in a car youcan find the instantaneous speed by looking at the speedometer. The instantaneous speed can change at any time andmy be changing continuously.

Average SpeedThe average speed is a measure of the total distance covered in some amount of time. In the car from the example abovethe average speed is the total distance for the trip divided by the total time for the trip, the car could have been travelingfaster or slower than that speed during parts of the trip, the average is only concerned with the total distance and thetotal time. Mathematically, you can find the average speed as

average speed =total distance covered

time interval

2.3 Velocity

In everyday language velocity is understood to mean the same as speed, but in physics there is an important distinction.Speed and velocity both describe the rate at which something is moving, but in addition to the rate velocity also includesthe direction of movement. You could describe the speed of a car as 60 km per hour, but the car’s velocity could be 60km per hour to the north, or to the south, or in any other direction so long as it is specified. You can change an object’svelocity without changing its speed by causing it to move in another direction but changing an object’s speed will alwaysalso change the velocity.

2.4 Acceleration

The rate at which an object’s velocity is changing is called its acceleration. It is a measure of how the velocity changeswith respect to time, so

acceleration =change in velocity

time interval

Be sure to realize that it is the change in velocity, not the velocity itself that involves acceleration. A person on a bicycleriding at a constant velocity of 30 km per hour has zero acceleration regardless of the time that they ride as long as thevelocity remains constant.

Acceleration in a negative direction will cause an object to slow down, this is often referred to as deceleration ornegative acceleration.

2.5 Free Fall: How Fast

When something is dropped from a height it will fall toward the earth because of gravity. If there is no air resistance(something that we will usually assume) then the object will fall with a constant acceleration and be in a state knownas free fall.

If you were to start a stopwatch at the instant the object was dropped the stopwatch would measure the elapsedtime for the fall.


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All objects in free fall will fall with the same acceleration, that is the acceleration caused by gravity and it is giventhe symbol g. A precise value would be g = 9.81 m

s2 but for most purposes we will make the math a bit easier and useg = 10 m

s2 . (Use g = 10 ms2 unless you are told otherwise.)

For an object starting at rest the relationship between the instantaneous velocity of the object (v) and the elapsedtime (t) is

v = gt

This equation will only work for an object dropped from rest, if the object is thrown upward or downward then theproblem has to be broken up into smaller pieces, taking advantage of the relationship between the speed of the object atthe same elevation as shown on page 18 of your textbook.

2.6 Free Fall: How FarIf an object is in free fall and dropped from rest then the relationship between the distance the object has fallen (d), theelapsed time (t), and the gravitational acceleration constant (g) is

d =12gt2

The mathematical models for free fall will also work for anything else moving in a straight line with constant accel-eration, all you need to do is replace the gravitational acceleration constant (g) with the acceleration of the object youare studying (a)

v = at d =12at2

2.7 Graphs of MotionFor an object starting at rest and moving with a constant acceleration a typical set of graphs of acceleration, velocity,and distance traveled would look like this when graphed with respect to time

Acceleration — a is a constant Velocity — v = at Distance — d = 12at

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At every point the value of the velocity graph is the slope of the distance graph, and the value of the acceleration graphis the slope of the velocity graph. Similarly, at every point the velocity graph is equal to the area under the accelerationgraph to that point and the displacement graph is equal to the area under the velocity graph to that point.

2.8 Air Resistance & Falling ObjectsWhen actual objects are dropped air resistance will cause different objects to fall in slightly different ways. For example,a brick will fall about the same way whether there is air resistance or not since it is heavy and compact but a piece ofpaper will tend to float downward. Wadding up the paper into a ball will cause it to fall faster by decreasing the areathat has to move through the air but it will still fall slower than a brick dropped at the same time.

2.9 How Fast, How Far, How Quickly How Fast ChangesVelocity is the rate of change of how far something has traveled, acceleration is the rate of change of the velocity or therate of change of the rate of change of the distance. This may sound confusing at first, if so back up and look at it apiece at a time as it was initially explained in the earlier parts of the chapter.


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Chapter 3 — Projectile Motion

3.1 Vector & Scalar QuantitiesA vector is something that requires both magnitude (size) and direction for a complete description. This could be howfar a rock has fallen (20 m downward), how quickly a car is accelerating (10 meters per second per second forward), howhard something is being pushed (with 35 newtons of force to the left), or any number of other quantities.

If something can be completely described with just magnitude alone then it is known as a scalar quantity. Examplesof scalars could be an elapsed time (22 seconds), a mass (15 kilograms) or a volume (0.30 cubic meters). Trying to adda direction to a scalar quantity (15 kilograms to the right) would not have any useful meaning in physics.

3.2 Velocity VectorsWhen vectors are drawn they are customarily drawn so that their length is proportional to the magnitude of the quantitythey represent. If a group of vectors are being added their sum (the resultant) can be found by drawing each of thevectors to scale and in the proper direction so that one vector starts where the previous one ends. If everything is drawnto scale then the resultant will just be another vector drawn from the beginning of the first vector to the end of the lastone. (See pages 30 and 31 of your textbook for examples of how this works)

3.3 Components of VectorsJust as any two (or more) vectors representing the same quantity (all velocity, all force, etc. — you can’t add a force toa velocity) can be added to find a single resultant vector you can also take any single vector and break it into two piecesthat are at right angles to each other. This is called resolution of the vector into components. This can be done bydrawing the vector to scale and at the proper angle, finding a rectangle that will just fit around it, and then measuringthe sides of the rectangle (as shown in your book on page 31) or you can do it mathematically using sine and cosine ifyou prefer.

3.4 Projectile MotionAny object that is shot, thrown, dropped, or otherwise winds up moving through the air (or even above the air in somecases) is known as a projectile. The horizontal and vertical motion of a projectile are independent of each other, thehorizontal motion is just motion with a constant velocity, vertically it is just free fall. Pages 33 and 34 of your textbookhave several examples of this type of motion.

3.5 Upwardly Launched ProjectilesBecause of gravity anything launched upward at an angle will follow a curved path and (usually, unless it is movingextremely fast) return to the earth. If there was no gravity then if you threw a softball it would travel forever in astraight line, because of gravity the softball will fall away from that straight line just as much as if it was dropped. (After1 second it will be 5 m below the line, after 2 seconds it will be 20 m below it, after 3 seconds 45 m, etc.)

The components of the velocity will change in very different ways as the projectile moves. The horizontal componentwill not change at all, the vertical component will be the same as for another object thrown straight up. You can takeadvantage of this to find how long something will be in the air from one of the components, then, using the time thatyou just found, find the other missing pieces of the problem.

An object thrown at 45 will travel farther than at any other angle, one thrown straight up will reach a highermaximum height.

For an object launched with an initial velocity vi at an angle of θ above the ground the distance it will travel is

d =v2

i sin(2θ)g

where g is just the familiar gravitational acceleration constant.

3.6 Fast-Moving Projectiles — SatellitesIf something is thrown extremely fast (faster than about 8 km per second) then it will never fall to the earth, insteadthe earth will curve away faster than the object will fall and it will go into orbit as a satellite.


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Chapter 4 — Newton’s First Law of Motion - Inertia

4.1 Aristotle on MotionThe Greek scientist Aristotle divided motion into natural motion (objects moving straight up or straight down, headingtoward their eventual resting place) and violent motion (motion imposed by an external cause moving an object awayfrom its resting place). According to Aristotle it is the nature of any object to come to rest and the object will do thison its own, it does not need any external influence for this to happen.

4.2 Copernicus and the Moving EarthCopernicus reasoned that the simplest way to explain astronomical observations was to assume that the the Earth andother planets moved around the sun instead of the common belief that the Earth was at the center off the universe.

4.3 Galileo on MotionOn of Galileo’s contributions to physics was the idea that a force is not necessary to keep an object moving.

A force is any push or pull on an object. Friction is the force that occurs when two objects rub against each otherand the small surface irregularities create a force that opposes the moving object(s). Galileo argued that only whenfriction is present is a force necessary to keep an object moving. The inertia of an object is a measure of its tendencyto keep moving as it is currently moving.

Galileo studied how objects moved rather than why. He showed that experiments instead of logic were the best testof ideas.

4.4 Newton’s Law of InertiaNewton’s first law, often called the inertia law states that every object continues in a state of rest, or of motion ina straight line at constant speed, unless it is compelled to change that state by forces exerted upon it.

Another way of saying this is that objects will continue to do whatever they are currently doing unless somethingcauses them to change. No force is required to maintain motion, a force is needed only to change it.

4.5 Mass — A Measure of InertiaMass is the amount of matter in an object or more specifically a measure of the inertia of an object that will resistchanges in motion.

The weight of an object is the gravitational force upon it, it can be found with the equationweight = mass × gravitational acceleration or weight = mg

4.6 Net ForceThe net force acting on an object is the sum of all of the forces acting on it. They may cancel each other out eithertotally or partially or they may combine to produce a force greater than any of the individual forces on the object.

4.7 Equilibrium — When Net Force Equals ZeroWhen the net force on an object is zero the object is in a state of equilibrium. This could be because all of the forcesare canceling each other out or it could be because there is actually no force on the object, either way the object willnot accelerate.

4.8 Vector Addition of ForcesForces are added as vectors just as displacement, velocity, and acceleration are. The same methods of vector additionwill work that you have used previously. Your textbook has several examples of vector addition of forces on pages 52–54.

4.9 The Moving Earth AgainAll measurements of motion are made relative to the observer. If you are in a moving car you can look at objects inthe car that appear stationary to you but in fact the are moving with you. The same thing happens with respect to theEarth. Since the Earth moves everyone on it will move with it and we will not notice that motion, everything moves onthe Earth as if the Earth was not moving.


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Chapter 5 — Newton’s Second Law of Motion — Force and Ac-celeration

5.1 Force Causes AccelerationIf the net force acting on an object is no zero the object will be accelerated, the acceleration of the object will be directlyproportional to the net force on the object — if the net force is doubled the acceleration will also be doubled.

5.2 Mass Resists AccelerationThe larger the mass of an object the smaller the acceleration that will be produced by a constant net force. Theexact relationship is that the acceleration is inversely proportional to the mass, another way of saying that is that theacceleration is proportional to one divided by the mass of the object.

5.3 Newton’s Second LawNewton’s second law states that the acceleration produced by a net force on an object is directly proportional to themagnitude of the net force, is in the same direction as the net force, and is inversely proportional to the mass of theobject.

Mathematically, if F is the net force, Newton’s second law can be expressed as

F = ma or a =F

m

5.4 FrictionFriction is a force that acts between any two objects in contact with each other to oppose their relative motion. The forceof friction depends on the surfaces in contact (rougher surfaces produce more friction) as well as the force pressing thesurfaces together (the more force pressing the surfaces together the more friction there will be). When frictional forcesare present they must be accounted for along with other applied forces to calculate the net force on objects.

5.5 Applying Force — PressureWhen two objects exert force on each other the pressure on the surface in contact can be found as

pressure =force

area of contact

The pressure can be increased by applying more force over the same area or by applying the same force over a smallerarea.

5.6 Free Fall ExplainedThe weight of an object is the force that causes an object to fall, and the weight is the product of the mass and thegravitational acceleration constant. Since F = ma the acceleration is equal to the force divided by the mass, this yields:

a =F

m=mg

m= mg

m= g

In the absence of air resistance or other frictional forces the acceleration of a falling object is equal to g, the gravitationalacceleration constant.

5.7 Falling and Air ResistanceAt low speeds air resistance (R) is very small and can usually be ignored for most objects, as the speed increases so doesthe air resistance. The heavier and smaller the object (such as a steel ball) the less air resistance will affect it, larger andlighter objects (such as a feather or a piece of paper) will be affected more because of the combination of a larger areaand a smaller mass.


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Chapter 6 — Newton’s Third Law of Motion - Action and Reaction

6.1 Forces and InteractionsA force is any push or pull on an object, but it also involves a second object to produce the force. This mutual action, orinteraction actually causes there to be a pair of forces — if a hammer applies a force to a nail the nail will also applya force back on the hammer.

6.2 Newton’s Third LawNewton’s third law states that whenever one object exerts a force on a second object, the second object exerts an equaland opposite force on the first object.

One force is called the action force, the other force is called the reaction force. Neither force can exist withoutthe other and they are equal in strength and opposite in direction. Newton’s third law is often expressed as “for everyaction there is an equal and opposite reaction”.

6.3 Identifying Action and ReactionWhen analyzing forces the action force may be apparent but the reaction force is often harder to see until you know whatto look for. Whatever the action force is, the reaction force always involves the same two objects. If the action force isthe Earth pulling down on you with your weight the reaction force would be you pulling up on the Earth with the sameforce.

6.4 Action and Reaction on Different MassesAction and reaction forces always act on a pair of objects. When a rifle is fired the action force is the rifle acting on thebullet, the reaction force is the bullet pushing back on the rifle. Since the bullet has a small mass and the rifle has alarge mass the bullet will have a larger acceleration.

6.5 Do Action and Reaction Forces Cancel?Since action and reaction forces each act upon different objects they can never cancel each other. In order to cancel eachother out forces must all act upon the same object.

6.6 The Horse-Cart ProblemSee the explanation and illustrations on pages 80–82 of your textbook for an excellent explanation of how action-reactionpairs work and how to handle multiple force pairs in one problem.

6.7 Action Equals ReactionIn order to apply a force to an object that object must be able to apply an equal and opposite force to you. The examplethat they use is trying to apply a 200 N punch to a piece of paper — when you apply even a small force to the paperit will easily accelerate out of your way from the applied force. Since it can’t stay in one place to apply a reaction forceyou cannot apply the larger action force.


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Chapter 7 — Momentum

7.1 MomentumMomentum is a vector quantity described by the product of an object’s mass times its velocity:

p = mv

Momentum is also known as inertia in motion. An object must be moving to have momentum, objects at rest have none.

7.2 Impulse Changes MomentumA force acting on an object for some amount of time produces an impusle which will change the object’s momentum.The impulse is defined as the product of the force and the time the force acts, so

impulse = Ft

Since impulse is also the change in momentum this then becomes

∆p = Ft

The same impulse can be delivered by a small force acting over a long time or a large force acting for a short time.If an object will be brought to a stop increasing the time during which the force acts will cause the force to be reduced,this is the principle behind padding, air bags, car bumpers, and many other safety devices.

7.3 BouncingWhen an object is brought to a stop the change in momentum is equal to the momentum that the object originally had.If the object bounces then the change in momentum will be twice the original momentum of the object, so an objectthat bounces will deliver twice the impulse of one that comes to a stop.

7.4 Conservation of MomentumWhen two objects interact the momentum lost by one object will be gained by the other. No momentum will be createdor destroyed, it will only be moved around between the objects. When momentum (or any other quantity, such as massor energy) acts like this we say that it is conserved. This leads to one of the principla laws of mechanics, the law ofconservation of momentum which states In the absence of an external force, the momentum of a system remainsunchanged. The external force being referred to in this case is one coming from an object not in the system.

7.5 CollisionsWhen two objects collide we will consider two cases, one where the two objects bounce off of each other (known as anelastic collision) and another where they stick together and travel as a single combined object (known as an inelasticcollision). In either case since momentum is conserved the momenta of the two objects before and after the collisionwill follow the relationship

m1v1i +m2v2i = m1v1f +m2v2f

7.6 Momentum VectorsSince momentum is a vector quantity it can act in any number of directions, not just in a straight line. The diagrams inyour textbook on page 98 help to illustrate what will happen in a collision when the two objects colliding do not bothmove in the same line.


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Chapter 8 — Energy

8.1 WorkWhen a force acts on an object over some distance (moving the object parallel to the direction of the force) the force issaid to do work (W ) on the object, the amount of work done is the product of the force times the distance over whichthe force was applied:

W = Fd

Work is measured in newtons times meters which are given the name joules (J).

8.2 PowerPower (P ) is defined to be the rate at which work is done, or the work done divided by the time interval during whichthe work was done:

P =W

t

Power is measured in joules divided by seconds which are given the name watts (W). One watt of power is what isrequired to convert one joule of energy every second.

8.3 Mechanical EnergyEnergy is the accumulation of work done in an object. If work is done in making an object move faster it has the abilityto do work on something else when it slows down (kinetic energy), if the work is done in raising it to some height thenit has the ability to do work as it is lowered back to its original position (potential energy). Just like work energy ismeasured in joules.

8.4 Potential EnergyPotential energy (PE ) is the energy stored in an object by lifting it to a higher position, the force required to lift theobject it its weight (mg) and the distance over which that force acts is the height (h), so the product of them is thepotential energy:

PE = mgh

Potential energy is a relative measurement, you choose the height that will be your zero level when you set up your frameof reference.

8.5 Kinetic EnergyKinetic energy (KE ) is the energy an object has because it is moving and it is equal to the work done in accelerating theobject to the speed at which it is traveling. The more mass or speed the object has the larger the kinetic energy:

KE =12mv2

8.6 Conservation of EnergyEnergy can be transferred from one object to another by one of them doing work on the other, it can also be transferredfrom one kind of energy to another within an object. This leads to the law of conservation of energy which statesthat energy cannot be created or destroyed. It can be transformed from one form into another, but the total amount ofenergy never changes. If anything exerts a force on an object in a system then the source of that force must also beincluded in the system for the law of conservation of energy to hold, otherwise energy could be moved into or out of thesystem.

For a single object moving without friction or other interactions the law of conservation of energy states that the sumof the kinetic and potential energies for the object will always be the same, so

PEi + KEi = PEf + KEf

or, by expanding those terms

mghi +12mv2

i = mghf +12mv2

f


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8.7 MachinesA machine is a device that is used to multiply forces or change the direction of forces. All machines are subject to thelaw of conservation of energy, no machine can produce more energy than is fed into it. In the ideal case no work will belost to friction so

Wout = Win

And since W = Fd this can becomeFoutdout = Findin

The amount that the machine multiplies the input force to get the output force is called the mechanical advantage(MA) of the machine:

MA =Fout

Fin

In the absence of friction the theoretical mechanical advantage (TMA) can also be expressed as the ratio of theinput distance to the output distance

TMA =din

dout

Some common machines are shown in your textbook on pages 112-115.

8.8 EfficiencyThe efficiency of a machine is the ratio of work output to work input, or

eff =Wout

Win

It can also be expressed as the ratio between the actual mechanical advantage (MA) and the theoretical mechanicaladvantage (TMA):

eff =MA

TMA

8.9 Energy for LifeLiving cells can also be regarded as machines, they take energy from either fuel or the sun and convert it to other forms.


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Chapter 9 — Circular Motion

9.1 Rotation and RevolutionIf an object rotates around an axis that passes through the object (an internal axis) then the motion is described asrotation, if the axis does not pass through the object then the motion is described as revolution. As an example, everyday the Earth rotates around its axis, once a year it revolves around the sun.

9.2 Rotational SpeedThe speed of a rotating object can be measured different ways, first the rotational speed (also known as angularspeed) measures how many rotations occur in a unit of time, for a rigid body (something that will not change shapeas you are studying it) every point on the object will have the same angular speed. The other type of speed is thetangential speed, that measures how fast a single point on the object is moving when measured along a straight linethat is tangent to the motion of that point. The farther an object is from the axis the larger the tangential speed willbe, a point on the axis will have a tangential speed of zero. The two speeds are related through the equation

tangential speed = radial distance × angular speed

9.3 Centripetal ForceIf an object is moving with no net force acting on it then Newton’s first law says it will move in a straight line. Tokeep an object moving in a circular path requires a constant force toward the center known as a centripetal force (acenter-seeking force). For an object of mass m moving in a circle of radius r with tangential speed v the centripetal force,F , is found by

F =mv2

r

9.4 Centripetal and Centrifugal ForcesIt it important to realize that the force causing circular motion is a centripetal force, not at force pulling away from theaxis of rotation (a centrifugal force or center-fleeing force). There is no centrifugal force.

9.5 Centrifugal Force in a Rotating ReferenceIf you are in some object that is moving in a circular path you might feel like you are experiencing a centrifugal forcebut in actuality there is no such force, what you are experiencing is the reaction force to the object pushing you towardthe center of the circle. No outward force exists on an object as a result of the object’s motion in a circle.

9.6 Simulated GravityIf a space station were constructed in the shape of a large wheel (as shown in the diagrams on page 131 of your textbook)and the wheel were rotated about the axis then anyone on the inside of the wheel would feel that they were being pulleddownward as if by gravity. This is caused by the centripetal force that keeps them moving in a circle, their reaction forcewill seem to pull them “down” toward the outside rim of the station.


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Chapter 10 — Center of Gravity

10.1 Center of GravityWhen studying physical objects it eventually becomes necessary to consider their size and shape. Up until now we havebeen looking at objects as if they could be represented by a single point. If an object has definite size and shape then itshould be possible to find a single point to represent the object, that point is the center of gravity of the object.

If an object is thrown through the air or slid across a frictionless table while it spins then it is the center of gravityof the object that will trace out a smooth curve. All of the other parts of the object will rotate around the center ofgravity when it spins.

10.2 Center of MassCenter of gravity is often also called center of mass, in most cases there is no difference although if an object is largeenough for the gravitational attraction to vary from one side to the other (such as a very tall building) then there maybe a very small difference between the two points. In this class we will often use the two terms interchangeably.

10.3 Locating the Center of GravityThe center of gravity of a uniform object is at the geometric center of the object, for a more complicated object it can befound by suspending the object from different points, tracing a line straight down from the point of suspension, repeatingthis, and then seeing where the lines cross — this point of intersection is the center of gravity of the object.

The center of gravity of an object may be located where no actual material of the object exists, for example the centerof gravity of a donut is in the middle of the hole.

10.4 TopplingEvery object that rests on a surface will contact that surface in one or more points, if you were to imagine a rubberband stretched around those points the area enclosed by that rubber band is the object’s supporting base. As long asthe center of gravity is above (or hanging below) that supporting base the object will remain in place. If the center ofgravity moves to no longer be over (or under) the supporting base then the object will fall over.

10.5 StabilityAn object with no net force acting on it can be either easy or difficult to tip over, depending on the size of the supportingbase of the object and how close the center of gravity is to the edge of the supporting base.

If an object is placed so that its center of gravity is directly over the edge of its supporting base then any smalldisplacement will start to lower the center of gravity and the object will tip over, this condition is known as unstableequilibrium.

If the object is placed so that its center of gravity is not on the edge of its supporting base then a small displacementwill start to raise the center of gravity and the object will return to the initial position when released, this is known adstable equilibrium.

A third condition also exists where the distance from the center of gravity to the outside of an object is the samefor any (or most) orientations of the object, in that case a small displacement will neither raise nor lower the center ofgravity and the object will remain in the new, displaced, position, this is known as neutral equilibrium.

10.6 Center of Gravity of PeopleA person’s center of gravity is about halfway between their front and back and a few centimeters below their navel ifthey are standing up straight, if they change their body position then the center of gravity will change as well. As youmove around you will generally shift your body to keep your center of mass somewhere over your supporting base.


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Chapter 11 — Rotational Mechanics

11.1 Torque

A force exerted some distance from the axis of rotation of an object is known as a torque. If a force is located on a linethat passes through the axis of rotation (or the center of mass of the object if there is no fixed axis) it will cause theobject to move in a straight line instead, no force passing through the axis of rotation can cause any torque.

The torque produced by a force is equal to lever arm (the distance from the axis of rotation to the point where theforce is acting) times the component of the force that is perpendicular to the lever arm, mathematically that would be

torque = lever arm × perpendicular force

The same torque can be produced by a large force with a short lever arm or a small force with a long lever arm.

11.2 Balanced Torques

If the sum of the torques in a clockwise direction is equal to the sum of the torques in the counterclockwise direction (anet torque of zero) then the object the torques are acting on is said to be in rotational equilibrium. Two people on anon-moving seesaw or a balanced scale would be examples of objects in rotational equilibrium.

See page 153 of your textbook for an example of how to use balanced torques to find unknown forces or distances.

11.3 Torque and Center of Gravity

Any force passing through the center of gravity of an object will tend to move the object instead of rotating it, if theforce instead passes some distance from the center of gravity then the object will experience a torque that will tend tocause the object to rotate as well as move.

11.4 Rotational Inertia

An object’s resistance to rotation is known as its rotational inertia, also known as moment of inertia. Just as anobject with larger mass is more difficult to start or stop moving than one with a small mass, an object with a largemoment of inertia is more difficult to start or stop rotating than one with a small moment of inertia.

The farther the mass of an object is from its axis of rotation the larger the object’s moment of inertia, an object withthe same mass closer to the axis of rotation will have a smaller moment of inertia. For an object of mass m with all ofits mass a distance r from the axis of rotation the moment of inertia I will be

I = mr2

Moments of inertia of more complicated objects are listed in your textbook on page 157.Since objects with large moments of inertia are more resistant to changing their rotational speed than those with

smaller moments of inertia if two or more objects are rolled down an inclined plane the first one to reach the bottom willbe the one with the smallest moment of inertia, a solid cylinder will beat a ring and a sphere will beat both the cylinderand the ring.

11.5 Rotational Inertia and Gymnastics

A human body can be considered to have three principal axes as pictured on page 159 of your textbook. They are thelongitudinal, transverse, and median axes. A body’s moment of inertia is least around the longitudinal axis and aboutequal around the other two.

11.6 Angular Momentum

Just as an object moving in a straight line has momentum equal to the product of its mass times its velocity an objectthat is rotating has angular momentum equal to the product of its moment of inertia times its angular speed. Unlikelinear momentum which is always the same for any mass and velocity the angular momentum of an object will varydepending on the arrangement of the object’s mass in addition to the mass and angular velocity.


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11.7 Conservation of Angular MomentumAngular momentum is conserved, a rotating object will always have the same amount unless the angular momentumis transferred to some other object. This, in combination with the dependence on the arrangement of mass will allowan object to change its rotational speed if the arrangement of mass changes, a notable example of this is standing on arotating platform while holding weights, if you start with your arms outstretched you will gain speed when you pull yourarms in despite your angular momentum remaining constant.


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Chapter 12 — Universal Gravitation

12.1 The Falling AppleNewton observed that falling objects (such as the infamous apple) were just one example of gravitational attraction,there is a gravitational attraction between every pair of objects no matter their size or distance.

12.2 The Falling MoonIn the absence of some external force the moon should continue to move in a straight line. Since it does not move in astraight line but instead orbits the earth there must be some force acting on it. This turns out to be the gravitationalattraction between the earth and the moon. Every second the moon falls toward the earth a very small amount relativeto where it would be if it kept traveling in a straight line, that distance that the moon falls is about 1.4 mm every second.

12.3 The Falling EarthJust as the moon falls around the earth, the earth and all of the other planets fall around the sun. If they ever stoppedmoving they would fall straight in toward the sun.

12.4 Newton’s Law of Universal GravitationNewton discovered that gravitational attraction is a universal force. Every bit of matter in the universe attracts everyother bit of matter, no matter how small they are or how far apart they may be. The exact force of gravitationalattraction, Fg, between any two objects of masses m1 and m2 separated by a distance r is

Fg = Gm1m2

r2

where G, the constant of universal gravitation, is found experimentally to be

G = 6.673× 10−11 N ·m2

kg2

12.5 Gravity and Distance: The inverse Square LawBecause the denominator of the equation above includes the square of the distance gravity is said to follow the inversesquare law. As the distance increases between two objects their force of attraction will decrease by the square of theamount the distance increased, for example doubling the distance will produce 1/4 of the force, tripling it will produce1/9 of the force, etc. There is a figure on page 176 in your textbook that helps to explain this phenomenon.

12.6 Universal GravitationAs discussed above, every particle of matter in the universe attracts every other particle. This has led to the discoveryof the planet Neptune because of the observed effects of its gravitational attraction on the planet Uranus, and the samething happened to lead to the discovery of the former planet Pluto.


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Chapter 13 — Gravitational Interactions

13.1 Gravitational FieldsGravity is a field force (a force that will act between two objects not in contact) and the strength of the gravitationalfield at any point is the acceleration that would be caused on an object at that point. At the surface of the earth thegravitational field strength is the familiar gravitational acceleration constant, g = 9.8 m

s2 .To find the gravitational field strength g at any point a distance r from the center of mass of a planet of mass m we

use a modification of the universal gravitation equation

g = Gm

r2

where G = 6.673× 10−11 N·m2

kg2

13.2 Gravitational Field Inside a PlanetThe equation above for the gravitational field strength assumes that all of the mass of the planet is on one side of you,if you burrow down into the planet then some of the planet’s mass will be on the other side of you, that will partiallycancel the force from the rest of the mass and reduce the effect. At the center of the planet the gravitational attractionfrom one side would be exactly balanced by the attraction from the other side and you would feel no net gravitationalforce.

13.3 Weight and WeightlessnessIf you are in an enclosure that is being accelerated you will feel a change in your apparent weight, think about beingin an elevator that is changing speed, if it’s starting to move upward you will feel heavier than normal, if it’s startingto move downward you will feel lighter than normal. If somehow the supporting cables were to break then you and theelevator car would both fall at the same rate and you would feel no weight relative to the elevator.

13.4 Ocean TidesSince the gravitational pull depends on the distance between the objects there is a difference in the gravitational pullfrom the moon to the near and far sides of the earth. For a solid object this would not be noticeable but for a liquidlike the water in the oceans it means that the water is pulled more on the side closer to the moon and less on the sideaway from the moon. This gives a tidal bulge on the side toward the moon (pulled there by the moon) and another oneon the side away from the moon (where there is a smaller pull from the moon and the oceans move away). There arealso similar tidal bulges produced by the sun but because the sun is much farther away they are less than half the sizeof those produced by the moon.

13.5 Tides and the Earth and AtmosphereThe earth itself is not a completely rigid object, it’s made up of a slightly flexible crust over a less solid inside. Becauseof this the earth’s surface itself is affected by the tidal forces. It’s not a noticeable effect since everything on the surfacemoves with the surface but it is measurable and has been found to be as much as 25 cm.

13.6 Black HolesStars are large accumulations of mostly gas that are constantly being affected by two forces — the gravitational forcepulling the gas in toward the center and the nuclear fusion in the core that pushes the gas back out. If the nuclear fusionweakens over time the gravitational force will pull the gas inward, and if there is enough gas there to pull inward theneventually the star will collapse inward onto itself and form what is known as a black hole. (Smaller stars will form whatis known as a back dwarf, effectively a cinder that is all that’s left of a burned-out star.)

The amount of mass present in the black hole is exactly the same as the mass in the star that collapsed so anythingthat was orbiting the star will orbit the black hole instead. The difference comes with the size of the black hole — sinceit is much smaller than the star it is now possible to get much closer to the center than was previously possible. Gettingcloser will greatly increase the gravitational force in that area and eventually the pull of gravity is so large that evenlight can’t get away from the center.


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Chapter 14 — Satellite Motion

14.1 Earth SatellitesA satellite is any object that falls around the earth instead of falling down into it. Since a projectile would drop about5 m every second and the earth’s curvature is such that you have to travel about 8000 m for the surface to curve away5 m if you are traveling at 8000 m per second near the surface of the earth then you will never hit the earth, you willalways fall around it instead.

As a practical matter if you are moving close to the surface of the earth then the friction of the atmosphere willquickly slow you down so most satellites stay about 150 km above the earth to avoid the atmosphere.

14.2 Circular OrbitsIf an object is in a circular orbit then it is always moving at the same speed and always at the same distance from theearth. There is no component of the gravitational force in the direction of the satellite’s motion so there can be noacceleration on the satellite.

The period of the satellite is the time that it takes for one complete orbit, if the satellite is close to the earth thenthe period is about 90 minutes.

A special kind of circular orbit is what is known as a geosynchronous or geostationary orbit, that is an orbit overthe equator with a period of 24 hours and will cause the satellite to always be above the same point on the earth. Thisis used for communications satellites and others that always need to be in the same position relative to the earth.

14.3 Elliptical OrbitsIf a satellite close to the earth has a speed higher than 8 km per second then it will overshoot a circular orbit and movefarther from the earth, trading off kinetic energy for potential energy as it slows down while increasing altitude. Thepath traced out will be an ellipse with the earth at one focus. (A circle is a special case of an ellipse with only one focus.)

The speed of the satellite will be smallest at the highest point (called the apogee) and largest at the lowest point(called the perigee).

14.4 Energy Conservation and Satellite MotionThe total energy (kinetic plus potential) of any satellite is constant. For an elliptical orbit one kind of energy is constantlybeing traded for the other, for a circular orbit both kinetic and potential energy remain constant, but in either case theirsum is always the same.

14.5 Escape SpeedThe escape speed of an object is the speed that something must travel to completely leave the earth (or another planetor star). For an object on the surface of the earth that speed is 11.2 km per second. Anything launched slower than thatspeed will eventually return to earth, anything faster will not. If the object starts from farther away then the escapespeed will be reduced, similarly if the planet has a smaller mass then the escape will also be reduced. There is a chartin your textbook on page 208 with the escape speeds from several bodies in the solar system.


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Chapter 25 — Vibrations and Waves

25.1 Vibration of a Pendulum

If you place a heavy object on the end of a light string and allow it to swing back and forth then you have a simplependulum. Galileo discovered that for small oscillations the time it takes for a pendulum to make one complete swingback and forth depends only on the length of the pendulum, not on the mass of the pendulum bob. This amount of timeto make one complete vibration is known as the period (T ) of the pendulum. A long pendulum will have a long period,a shorter pendulum will have a shorter period.

25.2 Wave Description

The back and forth motion of a pendulum is called simple harmonic motion. If the position of an object in simpleharmonic motion is graphed with respect to time the result will be a sine curve. A since curve has certain parts to itthat correspond to the parts of wave motion, these are the crests and troughs (the highest and lowest points on thewave), the amplitude (A, the distance from the center to the extreme points in the oscillation), and the wavelength(λ, the distance between any two adjacent corresponding parts of waves, such as one crest to the next or one trough tothe next). These are shown in this diagram:

sCrest

sTrough Wavelength (λ)

Amplitude

The number of vibrations that happen per second is called the frequency (f) of the oscillation, this is measured in Hertz(Hz) or inverse seconds (s−1). The relationship between period and frequency is an inverse one, so f = 1

T and T = 1f .

25.3 Wave Motion

A wave is a mechanism to transfer energy without transferring mass. You can think of a wave as an oscillation thatmoves from one location to another such as what happens to the surface of a pond when you throw a stone into it — theripples move outward in all directions from the initial disturbance.

25.4 Wave Speed

The speed of a wave (v) depends on the medium through which the wave moves. This course will not get into the specificsof how to determine wave speed but you should know that the more dense the medium the slower the speed will be andthe higher the pressure or tension the higher the speed will be.

Since the frequency of a wave is the number of full waves that pass a point in one second and the wavelength is howlong each of those waves is the speed of the wave is given by v = fλ.

25.5 Transverse Waves

Transverse waves are waves that vibrate perpendicular to the direction of wave travel like a wave in a stretched string.

25.6 Longitudinal Waves

Longitudinal waves or compression waves are waves that vibrate forward and backward along the direction of travellike sound waves traveling through air.

25.7 Interference

Unlike two physical objects, two waves are able to move through each other. Think about the result of dropping two stonesinto the water a small distance apart, the waves coming out from each of them will overlap and form an interferencepattern. If two waves line up so that a crest in one wave is on top of a crest in the other then the result will be what


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is known as constructive interference and the resulting wave will have a larger amplitude than either of the two thatare overlapping. If a crest lines up with a trough of the other wave then the result will be destructive interferenceand the resulting wave will be smaller, possibly zero. If two waves arrive at a point where the are in step with each otherwe say that the waves are in phase with each other, if they are out of step and cancel each other then we say that theyare out of phase with each other.

25.8 Standing WavesWhen a wave travels along through some medium and comes to a more dense (or rigid) medium then it will reflect backoff of that boundary. The incident wave will form an interference pattern with the reflected wave, and if the wavelengthof the wave is just right then what is known as a standing wave will be formed where the entire medium will appearto vibrate together. There are multiple wavelengths that will form a standing wave in the medium, the longest threewavelengths of them will produce patterns like this:s s

s s ss s s s

λ = 2L, 2 nodes, 1 antinode

λ = L, 3 nodes, 2 antinodes

λ = 23L, 4 nodes, 3 antinodes

Node Node Node NodeAntinode Antinode Antinode

In the bottom diagram the four points where there is no amplitude to the vibration are labeled as nodes and the pointsin between them where the vibration has maximum amplitude are marked as antinodes.

25.9 The Doppler EffectIf the source and detector of a sound are moving toward or away from each other the frequency of the sound observedwill change depending on the motion. If they move toward each other the frequency observed will be higher than thanthey are at rest, if they move away from each other then the observed frequency will be lower. This is known as theDoppler effect.

25.10 Bow WavesIf the source of a wave moves as fast as or faster than the speed of the wave then something different will be observed.Consider a boat causing ripples as it moves across a calm lake with some speed vs.

r r r rvs = 0 vs < vw vs = vw vs > vw

a b c

In the case where the source is not moving (vs = 0) a stationary observer at a will see the ripples pass their location atthe same frequency as they were emitted. If the source is moving slowly (vs < vw) a stationary observer at b will see theripples pass their location at a lower frequency than when they were emitted and one at c will see the ripples pass at ahigher frequency than when they were emitted.

In the case where the source is moving at a speed equal to the wave speed (vs = vw) then the waves will move forwardalong with the front of the boat, none will pass it, and the fronts of the waves will move forward in a straight line. If thesource is moving at a speed greater than the wave speed (vs > vw) then the boat will pass the fronts of the previouslyemitted waves and the fronts of the waves will move outward in a ’v’ shape that is commonly described as a wake.

25.11 Shock WavesThe three-dimensional version of a bow wave comes from a very fast plane moving through the air. Rather than a flat ’v’on the surface of a lake a plane moving faster than the speed of sound will produce a shock wave that follows behindthe plane in a cone.


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Chapter 26 — Sound

26.1 The Origin of Sound

All sounds are produced by the vibration of some object. In a musical instrument that is usually either a vibrating stringor reed or perhaps a moving column of air in part of the instrument. The production of sound is not limited to musicalinstruments, any vibrating object will cause similar vibrations in the air.

If the vibrations are between 20 Hz and 20 000 Hz then they are in the range that can be typically heard by aperson with normal hearing. Any waves below that range (less than 20 Hz) are called infrasonic, those above that range(greater than 20 000 Hz) are called ultrasonic. Humans cannot hear infrasonic or ultrasonic waves. The frequency thatwe perceive for those waves we do hear is called the pitch of the wave.

26.2 Sound in Air

The vibrating object that is the source of a sound will create a longitudinal wave composed of successive compressions(higher-pressure areas) and rarefactions (lower-pressure areas) that will travel outward from the source.

26.3 Media That Transmit Sound

We are used to sound traveling through the air but it will also travel through many other materials. In fact, sound willtravel better and faster through solid objects and liquids than through the air.

Like any other mechanical wave, sound cannot travel in absence of a medium. A vacuum will prevent sound frompassing since it contains nothing to vibrate.

26.4 Speed of Sound

Light travels much faster than sound, for this reason you will see a distant event (such as an explosion or lightning strike)before you hear the accompanying sound. Light travels fast enough that for most purposes its speed can be ignored andit can be assumed to travel instantly from one place to another.

The exact speed that sound waves travel in air varies with the temperature, barometric pressure, and humidity, about340 meters per second at room temperature is typical. They travel faster in warmer air and slower in cooler air.

In a more elastic material such as a solid or a liquid the atoms are packed more closely together and have a strongerforce restoring them to their original position once they are disturbed, this causes sound to travel much faster throughthose materials.

26.5 Loudness

The intensity of a sound wave is proportional to the square of its amplitude. The loudness of a sound will increase withthe intensity but it will not be a proportional relationship, instead it will increase with the logarithm of the intensity.Each time the the loudness seems to double the intensity will have increased by a factor of ten.

The units used to measure intensity are decibels (dB), an intensity of 0 dB corresponds to the threshold of hearingor the quietest sound that a person can typically hear. Every increase of 10 dB will cause the intensity to increase by afactor of 10, increasing by 20 dB will increase the intensity by a factor of 100.

At the same time, since the loudness that you will perceive doubles each time the intensity increases by a factor of10 the loudness will double with each 10 dB increase, a 20 dB increase will cause the sound to be four times as loud.

26.6 Forced Vibration

If a small object is vibrating then a soft sound will be heard, if the same object is placed against a larger flat surfacethen the larger surface act as a sounding board and will cause more air to vibrate so a louder sound will result. Thisphenomenon is known as forced vibration.

26.7 Natural Frequency

When an object that is made of an elastic material is struck it will vibrate at a frequency (or a set of frequencies) thatremains constant for that object. This is the object’s natural frquency. Consider a bell, if a bell is rung it will alwayssound the same, a different bell may have a different sound but it will also always be the same for that bell.


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26.8 ResonanceIf the frequency of forced vibration matches an object’s natural frequency then the resulting sound will increase inintentity and loundness, this is called resonance. If you recall the lab that we did regarding the speed of sound you werelooking for the length of an air column where resonance would occur, if it was too long or too short the sound would bebarely louder than the tuning fork but when the resonant length was found it would be much louder.

26.9 InterferenceSince sound wave share the properties of any other wave hey also show the effects of interference when two or morewaves reach the same location. If they arrive in phase the result will be constructive interference, meaning thatthe resulting wavefor will be larger than either of the waves that arrived. If the waves arrive out of phase then thedestructive interference that occurs will case the resulting wave to be smaller than either of the waves that areinterfering, perhaps even completely cancelled out. See the diagram at the top of page 397 in your textbook for examplesof constructive and destructive interference.

26.10 BeatsIf two sources emit sound waves that are close to one another in frequency but not exactly the same then a beat willbe observed. When the waves are added together the resulting sound oscillates between loud and soft. The number ofbeats heard per second is equal to the absolute value of the difference between the frequencies of the two sounds. Thediagram below shows one second of the sum of a 34 Hz wave and a 40 Hz wave yielding a 6 Hz beat frequency.

f = 40 Hz

f = 34 Hz

Sum

Harmonics and Standing WavesStrings and air-filled pipes can exhibit standing waves at certain frequencies. The lowest frequency able to form a standingwave is called the fundamental frequency (f1), those above it are harmonics of that frequency. The frequency ofeach harmonic frequency is equal to the fundamental frequency multiplied by the harmonic number (n), this forms aharmonic series. The fundamental frequency is also referred to as the first harmonic frequency.

Depending on the medium in which the standing waves are formed the characteristics of the waves will vary. A pipeopen at both ends has the same harmonic series as a vibrating string (producing all of the harmonics), while a pipeclosed at one end will only produce odd-numbered harmonics.

The harmonics that are present form a sound source will affect the timbre or sound quality of the resulting sound.

Standing Waves on a String or Open Pipe of Length L

n String Diagram Pipe Diagram Wavelength Frequency

1 λ1 = 2L f1 = v2L

2 λ2 = L f2 = vL = 2f1

3 λ3 = 23L f3 = 3v

2L = 3f1

n n = 1, 2, 3, . . . λn =2nL fn = n

v

2L= nf1

Standing Waves in a Pipe Closed at One Endn Diagram Wavelength Frequency

1 λ1 = 4L f1 = v4L

3 λ3 = 43L f3 = 3v

4L = 3f1

5 λ5 = 45L f5 = 5v

4L = 5f1

n n = 1, 3, 5, . . . λn =4nL fn = n

v

4L= nf1


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Chapter 27 — Light

27.1 Early Concepts of Light

Thousands of years ago light was thought to originate from the viewer’s eyes, later people learned that light originatedfrom an external source and came to the viewer’s eyes after interacting with other objects.

Once light was determined to have an external source there was still debate about whether light was a particle ora wave. In 1905 Einstein published a theory of how light was made up of particles of electromagnetic energy calledphotons, later it was discovered that even though they are particles the photons still show characteristics of waves.

27.2 The Speed of Light

Originally light was thought to travel instantaneously from one point to another. In the second half of the seventeenthcentury light was found to have a very large but finite speed and the first attempts were made to find that speed.

The experiment that finally gave the value currently accepted as the speed of light was done by Albert Michaelson in1880. He used a spinning octagonal mirror and a second mirror on a distant mountain to carefully determine the timethat light took to travel that distance.

The speed of light in a vacuum is slightly faster than in air, that speed (c) has been found to be

c = 3× 108 ms

= 300 000kms

Light takes about 8 minutes to travel from the sun to the earth.The distance that light travels in one year is called a light-year and is used to measure very long distances such as

between stars. The closest star to the sun, Alpha Centauri, is about 4 light-years away.

27.3 Electromagnetic Waves

The energy emitted by vibrating electric charges is partly electric and partly magnetic so it is referred to as an electro-magnetic wave. Light is a small part of the electromagnetic spectrum with wavelengths from 400 nm to 600 nm.Longer wavelengths are infrared (below red), microwaves, and radio waves. Shorter wavelengths are ultraviolet (aboveviolet), x-rays, and gamma rays. There is a diagram in your textbook on page 408 that shows the relationship betweenthe parts of the electromagnetic spectrum.

27.4 Light and Transparent Materials

When light travels through a transparent material it is successively absorbed and re-emitted by the atoms that itstrikes. This takes longer than the light would take to travel the same distance with no medium in the way, thereforelight is slowed in a transparent medium. Some media will pass electromagnetic waves of one frequency while blockingthose of another frequency, an example would be ordinary window glass which passes visible light while absorbing orreflecting ultraviolet.

27.5 Opaque Materials

Materials which absorb light without re-emitting it are called opaque. When light shines onto an opaque object theenergy will be transformed into heat and the object will become slightly warmer.

27.6 Shadows

A thin beam of light as we will work with it later is called a ray. A broader beam of light can be thought of as acollection of rays, perhaps all shining in the same direction, perhaps shining in different directions. Anything that thoserays would have reached if they were not blocked by some intervening object is said to be in that object’s shadow. If allof the rays are blocked from reaching a part of the surface then it is in the umbra of the object, if only some of the raysare blocked then it is in the penumbra. Your textbook has diagrams and an explanation of how this applies to eclipseson page 413.


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27.7 PolarizationLight can be caused to vibrate in one direction through polarization. Light traveling forward will normally vibrate upand down, left and right, and everything in between. When it is polarized it will all vibrate in a single direction. If thepolarized light encounters a second polarizing filter only the component of the originally polarized light that is in thedirection of the second filter will pass through. If the directions match then nearly all of the light will pass through, ifthe directions are at right angles to each other then nearly all of the light will be blocked.

Light can also be polarized by reflecting off of a surface, if light bounces off of a horizontal surface at a low angle itwill become horizontally polarized.

27.8 Polarized Light and 3-D ViewingPolarized light is used to produce 3-D images. Two projectors are used at one time, one projector has a horizontalpolarizing filter over the lens, the second one has a vertical polarizing filter. When the result is viewed through glassesthat have a similar combination of polarizing filters each eye will see the view from one of the projectors. If the imagesare set up correctly then the viewer will see a 3-D image.


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Chapter 28 — Color

28.1 The Color SpectrumIf you shine a beam of white light through a triangular prism you can see the light bend and split up by color into aspectrum. Red light will be bent least, violet will be bent most, and in between the order will be red, orange, yellow,green, blue, and violet.

If all of the colors of the spectrum were to be recombined the result will be white light again. If all of the light isremoved the result will be black.

28.2 Color By ReflectionWhen white light shines on an object some of the light will be absorbed and some will be reflected. Depending on thematerial the object is made of some wavelengths will be absorbed more than others, the remaining ones will be reflected.The colors of light that are reflected will be the ones that you see when you look at the object, they will determine thecolor that you see.

You can only see the colors reflect from an object that shine on the object to begin with, if you shine a single colorof light on an object, for example red, if the object reflects red light it will appear red, if not then it will be black. Noother colors are possible besides the ones shining on the object.

28.3 Color By TransmissionSimilar to how any object will absorb some wavelengths or light while reflecting others, if white light shines through atransparent object some wavelengths will be absorbed while others will pass through. The energy carried colors absorbedby the object will be turned into heat and will cause its temperature to slightly increase.

28.4 SunlightLight from the sun is a composite of all of the visible frequencies as well as many others that we cannot see. The brightestcolors coming from the sun are yellow and green with slightly less intensity as the frequencies move away from that point.Because humans developed in that light we are most sensitive to those frequencies. This increased sensitivity is whymany emergency vehicles such as fire engines are being painted that color.

28.5 Mixing Colored LightThe primary colors of light (the additive primary colors are red, blue, and green. All three together produce whitelight, red and green produce yellow, red and blue produce magenta, and blue and green produce cyan. There are coloreddiagrams to illustrate this on pages 426 and 427 of your textbook.

28.6 Complimentary ColorsComplimentary colors are pairs of colors that together include each primary color once, for example cyan (blue andgreen light) is complimentary to red. When two complimentary colors of light are mixed the result is white light.

28.7 Mixing Colored PigmentsPigments work by absorbing the complimentary colors to the ones that they reflect. A cyan pigment will absorb red lightand reflect blue and green.

If two pigments are mixed the resulting pigment will absorb all colors absorbed by either of the pigments, this isknown as mixing by subtraction. If yellow pigment (absorbs blue) is mixed with cyan pigment (absorbs red) the resultingpigment will absorb both blue and red and will reflect green light. A diagram of how this works and an example of apractical use of it is shown on pages 430 and 431 if your textbook. From the diagram and example you will see that thesubtractive primary colors are cyan, magenta, and yellow.

28.8 Why the Sky Is BlueAs light passes through microscopic particles in the upper atmosphere some of the light will be absorbed and re-emitted(scattered) by the particles it passes through. The shortest wavelengths of light (violet and blue) are scattered the most


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causing the sky to appear to be blue, the remaining colors of light (the longer wavelengths from green to red) will shinethrough causing the sun to appear to be yellow.

28.9 Why Sunsets Are RedWhen the sun is low in the sky the sunlight has a much longer path through the atmosphere to reach your eyes. Thismeans that more of the light (longer wavelengths) will be scattered leaving only red and orange light to reach the earth’ssurface.

28.10 Why Water Is Greenish BlueWater absorbs a very small amount of red light, when you see a small amount of water (a bottle or glass of water) youare unlikely to notice the coloration but when you see a large amount of water together the absence of red makes thewater look greenish blue or cyan.

28.11 The Atomic Color Code — Atomic SpectraEvery element emits its own characteristic set of colors of light when it is heated in a gaseous state. If the light comingfrom some element is viewed through a spectroscope then a pattern known as a line spectrum will be seen that isdifferent for every element. Several examples of line spectra are shown on page 438 of your textbook.

Primary Colors and Color Mixing

Red

Green BlueCyan

White

Yellow Magenta

(Light)Additive

Yellow

Cyan MagentaBlue

Black

Green Red

(Pigment)Subtractive

Additive and subtractive primary colorsColor Additive (mixing light) Subtractive (mixing pigment)red primary complimentary to cyangreen primary complimentary to magentablue primary complimentary to yellowcyan (blue green) complimentary to red primarymagenta (red blue) complimentary to green primaryyellow complimentary to blue primary


27

Chapter 29 — Reflection and Refraction

29.1 Reflection

When a wave reached the boundary between two media some or all of the wave will reflect back into the original medium.If all of the wave energy is reflected then it is said to be totally reflected, if some of it passes into the second mediumthen it is said to be partially reflected.

29.2 Refraction

If a wave is traveling in a one-dimensional medium (such as a pulse traveling along a spring) then the wave can only goforward and backward. When two or three dimensions are involved (we will stick to only two in this class) waves willreflect off of a boundary the same way a ball would bounce off of a wall — they will bounce off of the boundary at thesame angle they arrived but on the other side of an imaginary perpendicular line called the normal line. The anglesthat the wave arrives and departs at are measured from the normal line, the angle it arrives at is called the angle ofincidence and the departing angle is the angle of reflection. This leads to the mathematical relationship known asthe law of reflection:

angle of incidence = angle of reflection

29.3 Mirrors

If you look at an object in a plane mirror (an ordinary flat mirror) you will see the object some distance behind themirror. The rays of light reaching your eye do not actually start at the position of the image of the object so the imageis not actually there, but your eye can’t tell the difference when looking in the mirror. An image like this that you cansee but isn’t actually there is called a virtual image.

The virtual image in a plane mirror is always the same size as the object itself and will seem to be the same distancebehind the mirror as the object is in front of it.

29.4 Diffuse Reflection

A mirror is by design a very smooth surface but light can reflect from other surfaces as well. When light bounces off ofa rough surface the law of reflection is still obeyed but consider what happens if you try to catch a ball that’s going tobounce off of rough ground — depending on exactly where it strikes the ground the surface it hits could be tilted in anynumber of directions making the final direction depend on exactly where it strikes, a small difference in position couldmake a large difference in the direction. The same thing happens when light strikes a rough surface — the light ray willreflect perpendicular to the surface where it strikes but not all of the surface is facing the same direction. This results indiffuse reflection.

What is considered rough vs. smooth depends on the wavelength of the wave striking the surface, if the bumps onthe surface are small compared to the wavelength of the wave then it’s a smooth surface, if they’re large then it’s a roughsurface.

29.5 Reflection of Sound

Sound is a wave just like light or radio, the main difference is that sound has a much longer wavelength. When soundbounces back off of a distant object with a noticeable delay it’s called an echo, if it bounces back and forth in a roomit’s often called reverberation. Rooms like concert halls are designed to reduce some of the reverberation to improvethe quality of the sound for the audience.

29.6 Refraction

If a wave strikes a boundary between two different media at an angle then the wave will be bent or refracted as it passesinto the second medium. This is because the speed of the wave changes and the side of the wave that strikes the slowermedium first (or leaves it last) will drag a little and cause the wave to turn in that direction.

If you draw lines along the fronts of waves as they travel it can be helpful to visualize them, these are called wavefronts and will help to show how a wave changes its motion when changing media. See page 449 of your textbook forexamples.


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29.7 Refraction of SoundWaves do not need a hard boundary between media in order to refract, they will also bend if there is a gradual changein the speed such as in the atmosphere when a layer of cool air is under a layer of warm air, the sound moving faster inthe warm air will cause it to curve downward a bit causing distant sounds to seem to carry farther than they normallywould.

29.8 Refraction of LightWhen light rays move from a medium where light travels quickly such as air to one where it travels slower such as waterthe light rays will refract toward the normal line, if they go in the opposite direction then they will refract away fromthe normal line.

More precise results require the use of mathematics. The index of refraction (n) of a medium is defined to be:

index of refraction =speed of light in a vacuum

speed of light in the medium

For an index of refraction in an incident medium n and an index of refraction in a refracted medium n′ the relationshipbetween the angle of incidence (θ) and the angle of refraction (θ′) is given by Snell’s law:

n sin θ = n′ sin θ′

29.9 Atmospheric RefractionJust as sound waves will gradually bend when there is a small change in its speed in the air because of temperaturechanges light waves can be observed to do the same thing. If the air right above the ground is warm but the air abovethat is cooler then light waves will curve upward somewhat. If you look into the distance this can cause you to see amirage where you see what appears to be a reflecting surface such as water on the ground some distance away.

A similar effect will cause you to see the sun just before it actually rises over the horizon and just after it sets, thismakes a difference of about 5 minutes in the length of every day.

29.10 Dispersion in a PrismVarious factors cause blue and other short-wavelength light to travel through some media (such as glass and plastic) atdifferent speeds than longer-wavelength light such as red. Since all colors of light travel at nearly the same speed in air(and exactly the same speed in a vacuum) this means that the index of refraction for some materials varies for differentcolors of light. The colors with higher indices of refraction (the blue end of the spectrum) will refract a bit more thanthe ones with lower indices of refraction (the red end). The result of this will be a visible spectrum being produced whenlight shines through a prism.

29.11 The RainbowJust as a spectrum is produced when light shines through a prism it can also be produced when light shines throughother transparent media such as raindrops. Your textbook has an illustration on pages 455–466 of how this will form therainbow you are familiar with.

29.12 Total Internal ReflectionSince light traveling from a dense medium to a less dense one will speed up and refract away from the normal line therewill eventually be some angle of incidence where the angle of reflection is 90 or greater. The angle of incidence thatmakes the angle of refraction exactly 90 is called the critical angle. Your textbook illustrates this on page 458.


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Chapter 30 — Lenses

30.1 Converging and Diverging LensesA lens is a piece of glass or other transparent material that is specially shaped to change the path of light passingthrough it.

Lenses come in two types, converging lenses which cause parallel light rays to all come together to a single point(called the focal point) and diverging lenses which cause parallel light rays to spread out as if they came from a focalpoint on the other side of the lens. Converging lenses are thin at the edges and thicker in the middle, diverging lenseshave the opposite shape.

When drawing a diagram of an optical system start with a principal axis drawn perpendicular to and through thecenter of the lens. The focal points are marked on the principal axis one focal length from the lens. Your textbookillustrates this on page 464.

30.2 Image Formation by a LensIf the rays of light coming from an object actually come together to form the image it will be able to be projected on ascreen so it is a real image, if it can’t be projected because the rays do not actually converge at the image then it is avirtual image. Real images formed by a single lens are always upright, virtual ones are always inverted.

30.3 Constructing Images Through Ray DiagramsRay diagrams can also be drawn to locate and describe the image that will be formed by a lens. The usual rays to bedrawn are:

1. A ray that comes in parallel to the principal axis and goes out through the focal point.2. A ray that comes in through the focal point and goes out parallel to the principal axis.3. A ray that goes straight through the center of the lens

Each of these rays is labeled with its corresponding number in the ray diagrams below. We will use i to represent theimage distance, the distance from the image to the lens, o to represent the object distance, the distance from theobject to the lens, and f to represent the focal length of the lens. The focal points are labeled with F .

sF sF-

-

-

HHHH

HHHHHH

HHH

-*

HHHj

Object at ∞

Image

Converging lens, o = ∞, point image, i = f

1

3

1

sF

sFhhhhhhhhhhhhhhhhhhhhhh

XXXXXXXXXXXXXXX

-XXXXXXXXz

HHHHHH

HH

HHHj

-

Object

Image

Converging lens, o > 2f , real image, f < i < 2f

1

2

3

sF

sF-XXXzXXXXXXXXXXXXXXXXXXXX

HHHj

HHHH

HHHHHH

HHHH

HHHHHH

HHHj-

XXXXXXXXz

Object

ImageConverging lens, o = 2f , real image, i = 2f

1

2

3 sF

sFhhhhhhhhhhhhhhhhhhhhhh

HHHHH

HHH

-HHHHHj

-

XXXXXXXXXXXXXXX

XXXXXXXXz

Object

ImageConverging lens, f < o < 2f , real image, i > 2f

1

23

sF

sF-

HHHjHHH

HHHHHH

HHHHj

HHHHHH

HHHHHH

HHH

HHHHHH

HHHHHHH

HHHj

Object

No image

Converging lens, o = f , no image

1

3sF

sF

HHHHH

HHHHHH

HH

HHHHH

HHHHHj

XXXXXXXXXXXXX

XXXXXXXXz

-HHHXXXXX

Object

Image

Converging lens, o < f , virtual image

1

23


30

sF sF

HHHHH

HHHHHHHj

PPPPPP -

Image

Object

Diverging lens, virtual image

1

2

3

The relationship between the focal length, object distance, and image distance can also be found from the thin lensequation:

1o

+1i

=1f

30.4 Image Formation SummarizedAny image formed by a diverging lens will be upright and virtual.

An image formed by a converging lens will be inverted and real if the object is farther than the focal length from thelens, will be upright and virtual if the object is between the lens and the focal point, and no image will be formed if theimage is at the focal point.

30.5 Some Common Optical InstrumentsPages 471–473 of your textbook illustrate how lenses can be used to construct a camera, a telescope, a microscope, anda projector.

30.6 The EyePages 473 and 474 of your textbook discuss the structure of your eye. Of interest for this class is the converging lens atthe front of your eye which can change its focal length as needed (within a certain range) and the retina at the back ofthe eye. The lens will change focal length to bring an image of whatever is being looked at into focus on the retina.

30.7 Some Defects in VisionInstead of forming images on the retina as they are supposed to, some eyes will form an image behind the retina(farsighted) or in front of it (nearsighted). These can be corrected by placing corrective lenses in front of the eyes,either a converging lens to correct farsightedness or a diverging lens to correct nearsightedness.

30.8 Some Defects of LensesLenses have defects called aberrations. The most common are spherical aberration (light passing through the edgesof a spherical lens will focus at a different point than light passing close to the center) and chromatic aberration (lightof different colors will focus at different points.

Correcting spherical aberration requires that lenses be made parabolic in cross section rather than as a section of asphere, this is a considerably more expensive process.

To correct chromatic aberration requires that carefully chosen pairs (or more) of lenses be used so that their chromaticaberrations will cancel each other out.


31

Chapter 31 — Diffraction and Interference

31.1 Huygens’ PrincipleChristian Huygens developed what is now known as Huygens’ principle which states that every point on a wave frontacts as the source of a new wavelet that spreads out in a sphere from that point. This is illustrated on pages 481 and482 of your textbook.

31.2 DiffractionIf a wave bends because of anything other than reflection or refraction it is called diffraction. Whenever waves passthrough an opening or past an edge they will spread, the degree to which they spread will depend on the size of theopening and the wavelength of the waves.

When light passes through a large window very little diffraction will be seen, if it passes through a small slit then thediffraction will be much more pronounced. For a fixed opening the longer the wavelength of the wave the more diffractionthat will be observed. Long waves such as radio waves (especially AM radio) will diffract much more than shorter waveslike microwaves and visible light.

Diffraction also limits the ability of microscopes and other optical instruments to see small objects, once the size ofthe object is comparable to the wavelength of the light it will no longer produce a clear image.

31.3 InterferenceIf any two waves arrive in phase with each other they will produce a larger amplitude through constructive interference,if they are out of phase then they will cancel each other (in whole or in part) through destructive interference.

31.4 Young’s Interference ExperimentIn 1801 Thomas Young discovered that monochromatic light (light of all one wavelength) created an interferencepattern when it was allowed to shine through two small closely-spaced slits. The interference pattern and the waveinterference conditions that produce it are shown on pave 489 of your textbook.

A similar pattern is observed if the light instead shines on a diffraction grating (a glass or plastic plate with a largenumber of regularly-spaced grooves or slits). The pattern will be the same as if the light struck a single pair of grooveson the diffraction grating.

31.5 Single-Color Interference from Thin FilmsWhen monochromatic light reflects off of two surfaces that are almost (but not quite) parallel to each other a differentset of interference fringes will result. As shown on page 490 of your textbook, depending on the thickness at that pointeither constructive or destructive interference will result.

31.6 Iridescence of Thin FilmsThe position of thin-film interference fringes depends on the wavelength of the light illuminating the objects, as thewavelength of the light increases so does the width of and spacing between the fringes. As the various colors come intoconstructive and destructive interference the resulting colors observed at each point will be a combination of all of thecolors experiencing constructive interference there. As an example consider the bands of color you would observe lookingat a soap bubble or at gasoline spilled on a wet surface.


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31.7 Laser LightA beam of light with the same frequency (and thus wavelength), phase, and direction is said to be coherent. There isno destructive interference between the individual waves in a beam of coherent light.

A common source of coherent light is a laser (originally an acronym for “Light Amplification by Stimulated Emissionof Radiation”). A laser does not create energy, it just delivers its energy output in a way that makes it seem to be verybright. A laser commonly only outputs less than 1% of the energy put in to it.

31.8 The HologramA hologram is a three-dimensional picture of an object that contains an interference pattern generated by spreadinga laser beam so that half of it shines on the object and the other half shines on a piece of film. When the two beamscombine on the film they create an interference pattern. If a similar beam of light shines on the developed hologram thenthe waves will recreate the original waves that formed the interference pattern and the result will be a three-dimensionalview of the original object.


33

Chapter 32 — Electrostatics

32.1 Electrical Forces and ChargesPairs of electrical charges have a force that acts between them similar to the way that pairs of masses have a force actingbetween them. Unlike gravity which can only be an attractive force the electrical force can be either attractive orrepulsive. Two charges with opposite signs (one positive, one negative) will attract each other, two with the same sign(either both positive or both negative) will repel each other.

32.2 Conservation of ChargeThe principle of conservation of charge states that the total amount of charge, both positive and negative, is aconstant. Charge can be transferred from one object to another but it cannot be created or destroyed. If one object isgiven a positive charge something else must be given a negative charge of the same magnitude.

32.3 Coulomb’s LawThe electric force between two charges is given by Coulomb’s law which states:

F = kCq1q2d2

This is dependent on the Coulomb constant (kC):

kC = 9.0× 109 N ·m2

C2

The direction of the force will be toward the other charge if the two charges have opposite signs or away from the othercharge if they both have the same sign.

32.4 Conductors and InsulatorsSome materials will allow electrons to freely move through them, others prevent the easy movement of electrons. Ifelectrons are easy to move in a material that material is called a conductor, if the electrons are tightly bound toindividual atoms and do not easily move the material is called an insulator. Metals generally make very good conductors.Examples of insulators include plastic, rubber, and wood.

32.5 Charging by Friction and ContactWe can cause an object to become electrically charged though several methods, the one you are probably most familiarwith is charging by friction (or contact). When two dissimilar materials contact each other some of the electrons in onematerial will be drawn into the other material, this effect can be increased by rubbing the two surfaces together.

32.6 Charging by InductionIf a charged object is already available then it can be used to charge another object, even without touching the secondobject. Your textbook has diagrams on page 510 of charging by induction and by grounding.

32.7 Charge PolarizationWhen an insulator is brought near a charged object the charges in the insulator will rearrange slightly so that one side willhave a slight positive charge and the other will have a slight negative charge, causing the object to become electricallypolarized. This will not change the net charge of the insulator but it will be enough to allow a small or light object tobe lifted by a charged object. Your textbook shows examples of electrical polarization on pages 512 and 513.


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Chapter 33 — Electric Fields and Potential

33.1 Electric FieldsJust as a gravitational field exists around any object with mass that will exert a force on any other mass in that field,there is an electric field around any charged object that will exert a force on any other charged object placed in thatfield.

33.2 Electric Field LinesAn electric field around one or more charges can be represented by drawing lines to show the strength and direction ofthe field. They are illustrated on pages 519–520 of your textbook. When drawing field lines there are three main rules tofollow: 1) Field lines must begin on positive charges or at infinity and must terminate on negative charges or at infinity;2) The number of lines drawn leaving a positive charge or approaching a negative charge is proportional to the magnitudeof the charge; and 3) No two field lines from the same field can cross each other.

33.3 Electric ShieldingStatic charges on a conductor distribute themselves to be as far from each other as possible. This will cause the chargesto accumulate on the outside surface of the conductor with none of the unbalanced charges in the center. (This will alsocause the largest accumulation of charge to happen where the radius of curvature is smallest, the more pointed a part ofthe object is the more of the charge will accumulate there.)

As a result of the charge moving to the outer surface there will be no electric field on the inside of a conductor.

33.4 Electric Potential EnergyPotential energy exists anywhere work has been done to put an object into a new position (assuming that the energy hasnot been lost to friction). Just as it takes work to lift an object against Earth’s gravity it also takes work to move anobject in an electric field. The work done to move the object in the electric field will be stored in the object as electricpotential energy and can be recovered by allowing the object to return to its initial position.

33.5 Electric PotentialFinding electric potential energy requires that the charge on the object to be moved be known, at times it is useful tobe able to find the amount of electric potential energy an object would have without knowing in advance the object’scharge. This leads to the idea of finding the electric potential energy per unit charge for some location in an electric fieldwhich is known as electric potential and is defined as:

electric potential =electric potential energy

charge

Electric potential is measured in joules of energy per coulomb of charge, a quantity known as a volt (V). Because of theunites used to measure it electric potential is often known as voltage.

33.6 Electric Energy StorageElectrical energy can be stored in devices known as capacitors. A capacitor contains two plates or strips, one of whichwill take a positive charge and the other a negative. Placing the two plates close together with an insulator between themwill cause the opposite charges to hold each other in place until some way is given for the charges to equalize, usually byconnecting an electric circuit between the plates.

33.7 The Van de Graaff GeneratorA Van de Graaff generator is a device for building a very high voltage charge on a metal sphere which can then beused for various purposes. A diagram of one is in your textbook on page 527. It works by causing a small static chargeto be deposited on a moving belt in the base, the belt carries the charge to the top sphere where it is removed from thebelt on the inside of the sphere. The charge then travels to the surface of the sphere leaving the generator ready to movemore charge off of the belt. This will continue until there is enough charge on the sphere to either cause a long spark tosome other object or to ionize away into the air.


35

Chapter 34 — Electric Current

34.1 Flow of ChargeWhenever there is a potential difference between two points connected by a conductor some electric charge will flowthrough that conductor. The flow will continue as long as the potential difference is there. If the potential difference isbecause an object is charged then the flow will be fairly brief, if there’s something to keep the potential difference going(such as a battery or a generator) then the flow can continue for a long time.

34.2 Electric CurrentElectric current is the flow of electric charge. The charge is carried in a solid conductor by electrons that are able tomove through the conductor, in gases and liquids then there may be positive ions also carrying some of the charge.

Current is measured by the amount of charge that flows past a certain point, one coulomb of charge flowing in onesecond is called an ampere (A or amp).

Wires carrying a current do not have a charge themselves, as electrons enter one end of the wire the same numberleave the other end maintaining the wire’s neutral charge.

34.3 Voltage SourcesSomething that provides a potential difference between two points is known as a voltage source. Examples includebatteries, generators, and anything else that will cause a continuous flow of charges. You can think of a voltage sourceas a pump that makes the electrons move through the conductor.

34.4 Electric ResistanceElectric resistance is the property of conductors to resist the flow of electric charges. Longer or thinner conductorshave more resistance, shorter or thicker ones have less. The resistance also depends on the conductivity of the material,copper and aluminum have a high conductivity so wires made of those metals have a low resistance, other materials likenickel have a lower conductivity so they are only used in wires when a high resistance is needed.

Resistance is measured in ohms (Ω).

34.5 Ohm’s LawOhm’s law says that the current in a circuit (I) is equal to the voltage (V ) divided by the resistance (R), so:

I =V

R; R =

V

I; V = IR

An easy way to remember it is to draw it like this:

V

I R

If you put our finger on top of any one of the three quantities (V , I, or R) the remaining two will be in the right positionto tell you what to do with them to find the one you have covered. For example, to find current cover the I with yourfinger, what you then see is V

R which is equal to the current.

34.6 Ohm’s Law and Electric ShockWhen there is a potential difference across two parts of your body a small amount of charge will flow between them.Depending on the size of the current you may not notice it or you may be seriously injured.

If your skin is very dry it has a resistance of about 500 000 Ω but if it is soaked with salt water (or sweat) then thatcan drop to 100 Ω.

A current of about 0.001 A is the smallest that can be felt, about 0.005 A starts to be painful, and as little as 0.070 Acan be fatal if it passes through your heart.


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34.7 Direct Current and Alternating CurrentDirect current (DC) is a flow of charge that is always in the same direction. Currents caused by charged objects orbatteries are usually direct currents. Alternating current (AC) reverses direction many times per second. AC hasseveral advantages for transmitting power over long distances and converting one voltage to another when needed so itis often used to deliver electricity to homes.

34.8 Converting AC to DCA diode is a device that will allow a current in one direction but block it in another. Diodes can be used to pass onlya part of an AC source to change it into a pulsing DC source, other circuit elements can be added that will smooth outthe resulting DC output.

34.9 The Speed of Electrons in a CircuitWhen a circuit is completed electricity begins to flow almost immediately, this does not mean that the electrons thatleave the source make it through the circuit that quickly, only that they begin to move that fast. Individual electronstake much longer to make it around the circuit, moving at a drift speed of about 1 m every 3 hours.

34.10 The Source of Electrons in a CircuitThe electrons flowing through an electric circuit start out in the circuit itself. They are already present in the wires andother components, the potential difference causing the current just makes the electrons move, it does not supply them.

34.11 Electric PowerElectric power is the electrical energy that is converted to heat as it flows through a circuit. The amount of power isthe product of the current times the voltage (power = current × voltage) and is measured in watts.

Electrical energy is billed in kilowatt-hours, one of which is the amount of energy consumed in 1 hour at a rate of1 kilowatt or 3.6× 106 J.


37

Chapter 35 — Electric Circuits

35.1 A Battery and a BulbAn electric circuit forms a path between the two terminals of some potential difference. One of the simplest circuits isa battery and a light bulb connected between the terminals of the battery.

35.2 Electric CircuitsFor electricity to flow there must be a continuous path for it, if it is interrupted (such as by opening a switch) then thecurrent will stop. Multiple devices can be connected to the same circuit, if the electricity flows through one device afteranother then it is a series circuit, if the devices form multiple paths that the electricity can take then it is a parallelcircuit.

35.3 Series CircuitsThere are several characteristics of series circuits:

1. There is a single path through the circuit. If the current is interrupted at any point then all current will stop.

2. The total equivalent resistance of the circuit is the sum of the resistances of each element.

3. The current is equal to the potential difference divided by the total resistance. The current in each element is equalto the total current in the circuit.

4. The voltage drop across each element is equal to the current times the element’s resistance.

5. The total voltage across the circuit is equal to the sum of the voltage drops across each element.

35.4 Parallel CircuitsThere are several characteristics of parallel circuits:

1. Each element of a circuit connects the same two points so the voltage across each element is the same.

2. The amount of current in each element is inversely proportional to the resistance of the element.

3. The total current in the circuit is equal to the sum of the currents through each element.

4. As the number of parallel branches increases the total resistance of the decreases. The total resistance will be lessthan the resistance of any individual element.

35.5 Schematic DiagramsThe common symbols used in electrical schematic diagrams are shown on page 554 of your textbook.

35.6 Combining Resistors in a Compound CircuitIf two (or more) resistors are in series the total resistance is the sum of the individual resistances.

RT = R1 +R2 +R3 + · · ·

If two (or more) resistors are in parallel then the inverse of the total resistance is the sum of the inverses of the individualresistances.

1RT

=1R1

+1R2

+1R3

+ · · ·

If a circuit has resistors in both series and parallel then you just apply the same processes, starting with resistors thatare only in series or only in parallel. Once the first few resistors are combined then the resulting equivalent resistors arecombined with the remaining ones as shown on pages 555 and 556 of your textbook.

35.7 Parallel Circuits and OverloadingAs more devices are added to a parallel circuit the total current will increase, possibly to more than the wires can safelyhandle. Fuses and circuit breakers are used to limit the total current in a circuit by opening the circuit if they see morethan some preset limit.


38

Chapter 36 — Magnetism

36.1 Magnetic Poles

Just as electric charges produce electric forces magnetic poles produce magnetic forces. Each bit of a magnet has twopoles, a north-seeking pole that wants to point north and a south-seeking pole that wants to point south. If youhang a straight magnet from a string you will see it rotate so that the north-seeking pole points north if there are notother strong magnetic fields nearby. Also, just as like electric charges repel and unlike ones attract, like magnetic polesrepel and unlike ones attract.

Unlike electric charges which can be isolated (you can have a single positive or negative electric charge) magneticpoles always come in pairs. You cannot have a single north or south pole, only a set of one of each.

36.2 Magnetic Fields

A magnetic field is produced around any magnet and will affect other magnets placed nearby. If you sprinkle ironfilings on and around a magnet you will see them line up along magnetic field lines. This is because the small bits ofiron will act like miniature magnets when placed in the field and as such will line up along the stronger field.

36.3 The Nature of a Magnetic Field

In addition to the magnetic fields produced by magnets there are also magnetic fields produced any time electric chargesare in motion. (In fact, the magnetic fields produced by magnets are also caused by moving charges as electrons spin inplace, but we do not go into that level of detail in this class.)

36.4 Magnetic Domains

Materials such as iron have atoms that group together into clusters that all have the same magnetic orientation knownas magnetic domains. In unmagnetized iron the domains are randomly oriented and mostly cancel each other out, asthe iron starts to become magnetized the domains become aligned and start to constructively interfere. If the iron isstrongly magnetized then almost all of the domains will have the same orientation.

Once an object is magnetized by aligning its magnetic domains if the object is then broken each part of it will havetwo equally strong poles.

36.5 Electric Currents and Magnetic Fields

Since moving charges create magnetic fields and currents are charges in motion it follows that electric currents will havemagnetic fields around them as well. For a wire carrying current the magnetic field will wrap around the wire, if thewire is wrapped into a coil then the field will be strongest inside the coil. Photographs on page 569 of your textbookillustrate this.

36.6 Magnetic Forces on Moving Charged Particles

When a charged particle moves through a magnetic field in any direction other than parallel to the field the particle willfeel a force from the magnetic field and be deflected. (The force is greatest when the particle moves perpendicular to themagnetic field lines.) This property is used in televisions to guide a beam of electrons to strike the correct colored dotson the screen.

36.7 Magnetic Forces on Current-Carrying Wires

If the moving charges are in a wire then they will transfer the force to the wire causing the wire to move. If they arestrong enough they can evan cause a wire to jump out of a magnet.

36.8 Meters to Motors

Most (non-digital) electrical meters are made by wrapping a coil of wire around a magnet, when a current passes throughthe coil the magnet will experience a force and move causing a needle to move on a scale. By varying the circuit in whichthe meter is placed it can be set up to read current, voltage, or even resistance.


39

If the structure of a meter is enlarged then instead of just moving a needle the force on the magnet can be enough toproduce a useful torque. With a few appropriate design changes this forms the basis for electric motors.

36.9 The Earth’s Magnetic FieldThe earth and everything on it make up a very large magnet. Near the north pole is the magnetic north pole (which isactually a south-seeking pole), when you are far from the pole the two are close enough together that the difference is oflittle concern (in this area the difference in direction between true north and magnetic north is about 8) but as you getcloser to one of them it become more significant — there are several areas in the arctic where a magnetic compass willactually point south.

The exact cause of the earth’s magnetic field is unknown, current theories lean toward movements in the molten partof the earth’s interior and the charges that they carry with them as the cause.


40

Chapter 37 — Electromagnetic Induction

37.1 Electromagnetic Induction

When a magnet is moved in and out of (or through) a coil of wire it will induce a current in the coil though electro-magnetic induction. Moving the magnet faster or increasing the number of turns in the coil will increase the current,moving the wire the other direction will reverse the direction of the current.

37.2 Faraday’s Law

Electromagnetic induction is governed by Faraday’s law which states The induced voltage in a coil is proportional tothe product of the number of loops and the rate at which the magnetic field changes within those loops. The amount ofwork required to do this will depend on what the coil of wire is connected to.

37.3 Generators and Alternating Current

A generator is a device that continuously rotates a coil of wire inside a magnetic field, by Faraday’s law as the amountof the magnetic field passing through the coil changes (or if you prefer as the coil cuts through magnetic field lines) acurrent will be induced in the coil and it can be drawn off through electrical contacts on the shaft of the generator. Workis required to rotate the generator, that work done is converted (after frictional losses) into electrical energy.

37.4 Motor and Generator Comparison

An electric motor and a generator are effectively the same device, the difference is where the energy is put in and wherethe energy comes out. A motor converts electrical energy to mechanical energy, a generator converts mechanical energyto electrical energy.

37.5 Transformers

If two coils of wire are placed near each other and one of the coils is passing a varying electric current then the secondcoil will have an induced current because of the changing magnetic field around the first coil. If the two coils are placedaround the same iron core (as shown on page 584 of your textbook) this will create what is known as a transformer.A transformer can be used to increase or decrease the voltage coming from an alternating current source. The ratio bywhich the voltage will change is equal to the ratio of the number of turns on the output coil (the secondary coil) tothe number of turns on the input coil (the primary coil). The total power handled by the transformer is the same inboth coils. This can be expressed mathematically as

power = (voltage × current)primary = (voltage × current)secondary

37.6 Power Transmission

Electric utilities supply electricity in the form of alternating current, this allows it to be stepped up to high voltages tosend it long distances with little loss and then back down to 120 V to feed household circuits.

37.7 Induction of Electric and Magnetic Fields

Faraday’s law can be generalized to the case where there may not be a conductor in a changing electric field, the newversion states An electric field is created in any region of space in which a magnetic field is changing with time. Themagnitude of the created electric field is proportional to the rate at which the magnetic field changes. The direction of thecreated electric field is at right angles to the changing magnetic field. If a charge is within the changing magnetic field itwill experience a force from the induced electric field.

James Clark Maxwell promoted a counterpart to Faraday’s law to predict the result of changing an electric field: Amagnetic field is created in any region of space in which an electric field is changing with time. The magnitude of thecreated magnetic field is proportional to the rate at which the electric field changes. The direction of the created magneticfield is at right angles to the changing electric field.


41

37.8 Electromagnetic WavesAny time an electric current is present it will generate a magnetic field, but the changing magnetic field around a changingcurrent will also generate an electric field. If the two fields are arranged correctly the result will be an electromagneticwave as shown on page 590 of your textbook. As discovered by Maxwell, electromagnetic waves (such as light, radio,x-rays, etc.) are changing electric and magnetic fields that move at exactly the same speed and reinforce each other.Maxwell showed that the only speed at which the waves can move is the speed of light.


42

Variables and Notation

NotationNotation Description

~x Vectorx Scalar, or the magnitude of ~x|~x| The absolute value or magnitude of ~x∆x Change in x∑x Sum of all x

Πx Product of all xxi Initial value of xxf Final value of xx Unit vector in the direction of x

A −→ B A implies BA ∝ B A is proportional to B

UnitsSymbol Unit Quantity Composition

kg kilogram Mass SI base unitm meter Length SI base units second Time SI base unitA ampere Electric current SI base unitcd candela Luminous intensity SI base unitK kelvin Thermodynamic temperature SI base unit

mol mole Amount of substance SI base unitΩ ohm Resistance V

A or m2·kgs3·A2

C coulomb Charge A · sF farad Capacitance C

V or s4·A2

m2·kg

H henry Inductance V·sA or m2·kg

A2·s2Hz hertz Frequency s−1

J joule Energy N ·m or kg·m2

s2

N newton Force kg·ms2

rad radian Angle mm or 1

T tesla Magnetic field NA·m

V volt Electric potential JC or m2·kg

s3·AW watt Power J

s or kg·m2

s3

Wb weber Magnetic flux V · s or kg·ms2·A

ConstantsSymbol Name Established Value Value Usedε0 Permittivity of a vacuum 8.854 187 817× 10−12 C2

N·m2 8.85× 10−12 C2

N·m2

φ Golden ratio 1.618 033 988 749 894 848 20π Archimedes’ constant 3.141 592 653 589 793 238 46

g, ag Gravitational acceleration constant 9.81 ms2 (varies by location) 9.81 m

s2

c Speed of light in a vacuum 2.997 924 58× 108 ms (exact) 3.00× 108 m

se Natural logarithmic base 2.718 281 828 459 045 235 36e− Elementary charge 1.602 177 33× 1019 C 1.60× 1019 CG Gravitational constant 6.672 59× 10−11 N·m2

kg2 6.67× 10−11 N·m2

kg2

kC Coulomb’s constant 8.987 551 788× 109 N·m2

C2 8.99× 109 N·m2

C2

NA Avogadro’s constant 6.022 141 5× 1023 mol−1


43

VariablesVariable Description Units

α Angular acceleration rads2

θ Angular position radθc Critical angle o (degrees)θi Incident angle o (degrees)θr Refracted angle o (degrees)θ′ Reflected angle o (degrees)∆θ Angular displacement radτ Torque N ·mω Angular speed rad

sµ Coefficient of friction (unitless)µk Coefficient of kinetic friction (unitless)µs Coefficient of static friction (unitless)~a Acceleration m

s2

~ac Centripetal acceleration ms2

~ag Gravitational acceleration ms2

~at Tangential acceleration ms2

~ax Acceleration in the x direction ms2

~ay Acceleration in the y direction ms2

A Area m2

B Magnetic field strength TC Capacitance F~d Displacement m

d sin θ lever arm m~dx or ∆x Displacement in the x direction m~dy or ∆y Displacement in the y direction m

E Electric field strength NC

f Focal length m~F Force N~Fc Centripetal force N

~Felectric Electrical force N~Fg Gravitational force N~Fk Kinetic frictional force N

~Fmagnetic Magnetic force N~Fn Normal force N~Fs Static frictional force N~F∆t Impulse N · s or kg·m

s~g Gravitational acceleration m

s2

Variable Description Unitsh Object height mh′ Image height mI Current AI Moment of inertia kg ·m2

KE or K Kinetic energy JKE rot Rotational kinetic energy JL Angular Momentum kg·m2

sL Self-inductance Hm Mass kgM Magnification (unitless)M Mutual inductance HMA Mechanical Advantage (unitless)ME Mechanical Energy Jn Index of refraction (unitless)p Object distance m~p Momentum kg·m

sP Power W

PE or U Potential Energy JPE elastic Elastic potential energy JPE electric Electrical potential energy J

PE g Gravitational potential energy Jq Image distance m

q or Q Charge CQ Heat, Entropy JR Radius of curvature mR Resistance Ωs Arc length mt Time s

∆t Time interval s~v Velocity m

svt Tangential speed m

s~vx Velocity in the x direction m

s~vy Velocity in the y direction m

sV Electric potential V

∆V Electric potential difference VV Volume m3

W Work Jx or y Position m

Physics

2007–2008M

r.Strong

44

SI PrefixesPrefix Multiple Abbrev.yocto- 10−24 yzepto- 10−21 zatto- 10−18 afemto- 10−15 fpico- 10−12 pnano- 10−9 nmicro- 10−6 µmilli- 10−3 mcenti- 10−2 cdeci- 10−1 ddeka- 101 dahecto- 102 hkilo- 103 kmega- 106 Mgiga- 109 Gtera- 1012 Tpeta- 1015 Pexa- 1018 Ezetta- 1021 Zyotta- 1024 Y

Greek AlphabetName Majuscule MinusculeAlpha A αBeta B βGamma Γ γDelta ∆ δEpsilon E ε or εZeta Z ζEta H ηTheta Θ θ or ϑIota I ιKappa K κLambda Λ λMu M µNu N νXi Ξ ξOmicron O oPi Π π or $Rho P ρ or %Sigma Σ σ or ςTau T τUpsilon Υ υPhi Φ φ or ϕChi X χPsi Ψ ψOmega Ω ω

Astronomical DataSymbol Object Mean Radius Mass Mean Orbit Radius Orbit PeriodÁ Moon 1.74× 106 m 7.36× 1022 kg 3.84× 108 m 2.36× 106 sÀ Sun 6.96× 108 m 1.99× 1030 kg — —Â Mercury 2.43× 106 m 3.18× 1023 kg 5.79× 1010 m 7.60× 106 sÃ Venus 6.06× 106 m 4.88× 1024 kg 1.08× 1011 m 1.94× 107 sÊ Earth 6.37× 106 m 5.98× 1024 kg 1.496× 1011 m 3.156× 107 sÄ Mars 3.37× 106 m 6.42× 1023 kg 2.28× 1011 m 5.94× 107 s

Ceres1 4.71× 105 m 9.5× 1020 kg 4.14× 1011 m 1.45× 108 sÅ Jupiter 6.99× 107 m 1.90× 1027 kg 7.78× 1011 m 3.74× 108 sÆ Saturn 5.85× 107 m 5.68× 1026 kg 1.43× 1012 m 9.35× 108 sÇ Uranus 2.33× 107 m 8.68× 1025 kg 2.87× 1012 m 2.64× 109 sÈ Neptune 2.21× 107 m 1.03× 1026 kg 4.50× 1012 m 5.22× 109 sÉ Pluto1 1.15× 106 m 1.31× 1022 kg 5.91× 1012 m 7.82× 109 s

Eris21 2.4× 106 m 1.5× 1022 kg 1.01× 1013 m 1.75× 1010 s

1Ceres, Pluto, and Eris are classified as “Dwarf Planets” by the IAU2Eris was formerly known as 2003 UB313


45

Mathematics

Algebra

Fundamental properties of algebraa+ b = b+ a Commutative law for addition(a+ b) + c = a+ (b+ c) Associative law for additiona+ 0 = 0 + a = a Identity law for additiona+ (−a) = (−a) + a = 0 Inverse law for additionab = ba Commutative law for multiplication(ab)c = a(bc) Associative law for multiplication(a)(1) = (1)(a) = a Identity law for multiplicationa 1

a = 1aa = 1 Inverse law for multiplication

a(b+ c) = ab+ ac Distributive law

Exponentsanam = an+m (ab)n = anbn 0n = 0an/am = an−m (a/b)n = an/bn a0 = 1(an)m = a(mn) 00 = 1 (by definition)

Logarithmsx = ay −→ y = loga x

loga(xy) = loga x+ loga y

loga

(x

y

)= loga x− loga y

loga(xn) = n loga x

loga

(1x

)= − loga x

loga x =logb x

logb a= (logb x)(loga b)

Binomial Expansions(a+ b)2 = a2 + 2ab+ b2

(a+ b)3 = a3 + 3a2b+ 3ab2 + b3

(a+ b)n =n∑

i=0

n!i!(n− i)!

aibn−i

Quadratic formulaFor equations of the form ax2 + bx+ c = 0 the solutions are:

x =−b±

√b2 − 4ac

2a

GeometryShape Area VolumeTriangle A = 1

2bh —Rectangle A = lw —Rectangular Prism A = 2(lw + lh+ hw) V = lwhCircle A = πr2 —Sphere A = 4πr2 V = 4

3πr3

Cylinder A = 2πrh+ 2πr2 V = πr2h

Cone A = πr√r2 + h2 + πr2 V = 1

3πr2h


46

Trigonometry

In physics only a small subset of what is covered in a trigonometry class is likely to be used, in particular sine, cosine,and tangent are useful, as are their inverse functions. As a reminder, the relationships between those functions and thesides of a right triangle are summarized as follows:

θ

opp

adj

hypsin θ =

opp

hypcos θ =

adj

hyp

tan θ =opp

adj=

sin θcos θ

The inverse functions are only defined over a limited range. The tan−1 x function will yield a value in the range−90 < θ < 90, sin−1 x will be in −90 ≤ θ ≤ 90, and cos−1 x will yield one in 0 ≤ θ ≤ 180. Care must be taken toensure that the result given by a calculator is in the correct quadrant, if it is not then an appropriate correction must bemade.

Degrees 0o 30o 45o 60o 90o 120o 135o 150o 180o

Radians 0 π6

π4

π3

π2

2π3

3π4

5π6 π

sin 0 12

√2

2

√3

2 1√

32

√2

212 0

cos 1√

32

√2

212 0 − 1

2 −√

22 −

√3

2 -1tan 0

√3

3 1√

3 ∞ −√

3 -1 −√

33 0

sin2 θ + cos2 θ = 1

2 sin θ cos θ = sin(2θ)

HHH

HHH

A

B

C

a

b

c

a2 = b2 + c2 − 2bc cosAb2 = a2 + c2 − 2ac cosBc2 = a2 + b2 − 2ab cosC

a

sinA=

b

sinB=

c

sinC

let s = 12 (a+ b+ c)

Area=√s(s− a)(s− b)(s− c)

Area= 12bc sinA = 1

2ac sinB = 12ab sinC

Vectors

A vector is a quantity with both magnitude and direction, such as displacement or velocity. Your textbook indicates avector in bold-face type as V and in class we have been using ~V . Both notations are equivalent.

A scalar is a quantity with only magnitude. This can either be a quantity that is directionless such as time or mass,or it can be the magnitude of a vector quantity such as speed or distance traveled. Your textbook indicates a scalar initalic type as V , in class we have not done anything to distinguish a scalar quantity. The magnitude of ~V is written asV or |~V |.

A unit vector is a vector with magnitude 1 (a dimensionless constant) pointing in some significant direction. A unitvector pointing in the direction of the vector ~V is indicated as V and would commonly be called V -hat. Any vector canbe normalized into a unit vector by dividing it by its magnitude, giving V = ~V

V . Three special unit vectors, ı, , andk are introduced with chapter 3. They point in the directions of the positive x, y, and z axes, respectively (as shownbelow).


47

6

?

-

y+

−

x+−

z+

−6

-

ı

k

Vectors can be added to other vectors of the same dimension (i.e. a velocity vector can be added to another velocityvector, but not to a force vector). The sum of all vectors to be added is called the resultant and is equivalent to all ofthe vectors combined.

Multiplying VectorsAny vector can be multiplied by any scalar, this has the effect of changing the magnitude of the vector but not itsdirection (with the exception that multiplying a vector by a negative scalar will reverse the direction of the vector). Asan example, multiplying a vector ~V by several scalars would give:

- - - ~V 2~V .5~V −1~V

In addition to scalar multiplication there are also two ways to multiply vectors by other vectors. They will not bedirectly used in class but being familiar with them may help to understand how some physics equations are derived. Thefirst, the dot product of vectors ~V1 and ~V2, represented as ~V1 · ~V2 measures the tendency of the two vectors to point inthe same direction. If the angle between the two vectors is θ the dot product yields a scalar value as

~V1 · ~V2 = V1V2 cos θ

The second method of multiplying two vectors, the cross product, (represented as ~V1 × ~V2) measures the tendencyof vectors to be perpendicular to each other. It yields a third vector perpendicular to the two original vectors withmagnitude

|~V1 × ~V2| = V1V2 sin θ

Adding Vectors GraphicallyThe sum of any number of vectors can be found by drawing them head-to-tail to scale and in proper orientation thendrawing the resultant vector from the tail of the first vector to the point of the last one. If the vectors were drawnaccurately then the magnitude and direction of the resultant can be measured with a ruler and protractor. In theexample below the vectors ~V1, ~V2, and ~V3 are added to yield ~Vr

-

AAAAAAU

~V1

~V2 ~V3

~V1, ~V2, ~V3

-AAAAAAU

~V1

~V2

~V3

~V1 + ~V2 + ~V3

-AAAAAAU

~V1

~V2

~V3

-~Vr

~Vr = ~V1 + ~V2 + ~V3

Adding Parallel VectorsAny number of parallel vectors can be directly added by adding their magnitudes if one direction is chosen as positiveand vectors in the opposite direction are assigned a negative magnitude for the purposes of adding them. The sum ofthe magnitudes will be the magnitude of the resultant vector in the positive direction, if the sum is negative then theresultant will point in the negative direction.

Adding Perpendicular VectorsPerpendicular vectors can be added by drawing them as a right triangle and then finding the magnitude and direction ofthe hypotenuse (the resultant) through trigonometry and the Pythagorean theorem. If ~Vr = ~Vx + ~Vy and ~Vx ⊥ ~Vy thenit works as follows:


48

-~Vx

6

~Vy

-~Vx

6

~Vy

~Vr

θ

Since the two vectors to be added and the resultant form a right triangle with the resultant as the hypotenuse thePythagorean theorem applies giving

Vr = |~Vr| =√V 2

x + V 2y

The angle θ can be found by taking the inverse tangent of the ratio between the magnitudes of the vertical and horizontalvectors, thus

θ = tan−1 Vy

Vx

As was mentioned above, care must be taken to ensure that the angle given by the calculator is in the appropriatequadrant for the problem, this can be checked by looking at the diagram drawn to solve the problem and verifying thatthe answer points in the direction expected, if not then make an appropriate correction.

Resolving a Vector Into Components

Just two perpendicular vectors can be added to find a single resultant, any single vector ~V can be resolved into twoperpendicular component vectors ~Vx and ~Vy so that ~V = ~Vx + ~Vy.

~V

θ ~Vx

~Vy

~V

θ

-

6

As the vector and its components can be drawn as a right triangle the ratios of the sides can be found with trigonometry.Since sin θ = Vy

V and cos θ = Vx

V it follows that Vx = V cos θ and Vy = V sin θ or in a vector form, ~Vx = V cos θı and~Vy = V sin θ. (This is actually an application of the dot product, ~Vx = (~V · ı)ı and ~Vy = (~V · ), but it is not necessaryto know that for this class)

Adding Any Two Vectors AlgebraicallyOnly vectors with the same direction can be directly added, so if vectors pointing in multiple directions must be addedthey must first be broken down into their components, then the components are added and resolved into a single resultantvector — if in two dimensions ~Vr = ~V1 + ~V2 then

*

~V1

~V2

~Vr

~Vr = ~V1 + ~V2

*

-

-6

6

~V1

~V2

~Vr

~V1x

~V2x

~V1y

~V2y

~V1 = ~V1x + ~V1y

~V2 = ~V2x + ~V2y

~Vr = (~V1x + ~V1y) + (~V2x + ~V2y)

*

- -

6

6

~V1

~V2

~Vr

~V1x~V2x

~V1y

~V2y

~Vr = (~V1x + ~V2x) + (~V1y + ~V2y)Once the sums of the component vectors in each direction have been found the resultant can be found from them justas an other perpendicular vectors may be added. Since from the last figure ~Vr = (~V1x + ~V2x) + (~V1y + ~V2y) and it waspreviously established that ~Vx = V cos θı and ~Vy = V sin θ it follows that

~Vr = (V1 cos θ1 + V2 cos θ2) ı+ (V1 sin θ1 + V2 sin θ2)


49

andVr =

√(V1x + V2x)2 + (V1y + V2y)2 =

√(V1 cos θ1 + V2 cos θ2)

2 + (V1 sin θ1 + V2 sin θ2)2

with the direction of the resultant vector ~Vr, θr, being found with

θr = tan−1 V1y + V2y

V1x + V2x= tan−1 V1 sin θ1 + V2 sin θ2

V1 cos θ1 + V2 cos θ2

Adding Any Number of Vectors Algebraically

For a total of n vectors ~Vi being added with magnitudes Vi and directions θi the magnitude and direction are:

*

- - -

6

6

6

~V1

~V2

~V3

~Vr

~V1x~V2x

~V3x

~V1y

~V2y

~V3y

~Vr =

(n∑

i=1

Vi cos θi

)ı+

(n∑

i=1

Vi sin θi

)

Vr =

√√√√( n∑i=1

Vi cos θi

)2

+

(n∑

i=1

Vi sin θi

)2

θr = tan−1

n∑i=1

Vi sin θi

n∑i=1

Vi cos θi

(The figure shows only three vectors but this method will work with any number of them so long as proper care is takento ensure that all angles are measured the same way and that the resultant direction is in the proper quadrant.)

CalculusAlthough this course is based on algebra and not calculus, it is sometimes useful to know some of the properties ofderivatives and integrals. If you have not yet learned calculus it is safe to skip this section.

The derivative of a function f(t) with respect to t is a function equal to the slope of a graph of f(t) vs. t at everypoint, assuming that slope exists. There are several ways to indicate that derivative, including:

d

dtf(t)

f ′(t)

f(t)

The first one from that list is unambiguous as to the independent variable, the others assume that there is only onevariable, or in the case of the third one that the derivative is taken with respect to time. Some common derivatives are:

d

dttn = ntn−1

d

dtsin t = cos t

d

dtcos t = − sin t

d

dtet = et

If the function is a compound function then there are a few useful rules to find its derivative:

d

dtcf(t) = c

d

dtf(t) c constant for all t

d

dt[f(t) + g(t)] =

d

dtf(t) +

d

dtg(t)


50

d

dtf(u) =

d

dtud

duf(u)

The integral, or antiderivative of a function f(t) with respect to t is a function equal to the the area under a graph off(t) vs. t at every point plus a constant, assuming that f(t) is continuous. Integrals can be either indefinite or definite.An indefinite integral is indicated as: ∫

f(t)dt

while a definite integral would be indicated as ∫ b

a

f(t)dt

to show that it is evaluated from t = a to t = b, as in:∫ b

a

f(t)dt =∫f(t)dt|t=b

t=a =∫f(t)dt|t=b −

∫f(t)dt|t=a

Some common integrals are: ∫tndt =

tn+1

n+ 1+ C n 6= −1∫

t−1dt = ln t+ C∫sin tdt = − cos t+ C∫cos tdt = sin t+ C∫etdt = et + C

There are rules to reduce integrals of some compund functions to simpler forms (there is no general rule to reduce theintegral of the product of two functions):∫

cf(t)dt = c

∫f(t)dt c constant for all t

∫[f(t) + g(t)]dt =

∫f(t)dt+

∫g(t)dt

Some functions are difficult to work with in their normal forms, but once converted to their series expansion can bemanipulated easily. Some common series expansions are:

sinx = x− x3

3!+x5

5!− x7

7!+ · · · =

∞∑i=0

(−1)i x2i+1

(2i+ 1)!

cosx = 1− x2

2!+x4

4!− x6

6!+ · · · =

∞∑i=0

(−1)i x2i

(2i)!

ex = 1 + x+x2

2!+x3

3!+x4

4!+ · · · =

∞∑i=0

xi

(i)!


51

Data Analysis

Comparative MeasuresPercent ErrorWhen experiments are conducted where there is a calculated result (or other known value) against which the observedvalues will be compared the percent error between the observed and calculated values can be found with

percent error =∣∣∣∣observed value − accepted value

accepted value

∣∣∣∣× 100%

Percent DifferenceWhen experiments involve a comparison between two experimentally determined values where neither is regarded ascorrect then rather than the percent error the percent difference can be calculated. Instead of dividing by the acceptedvalue the difference is divided by the mean of the values being compared. For any two values, a and b, the percentdifference is

percent difference =∣∣∣∣ a− b

12 (a+ b)

∣∣∣∣× 100%

Dimensional Analysis and Units on ConstantsConsider a mass vibrating on the end of a spring. The equation to find the mass from the frequency from the graphmight look something like

y =3.658x2

− 1.62

This isn’t what we’re looking for, a first pass would be to change the x and y variables into f and m for frequency andmass, this would then give

m =3.658f2

− 1.62

This is progress but there’s still a long way to go. Mass is measured in kg and frequency is measured in s−1, since thesevariables represent the entire measurement (including the units) and not just the numeric portion they do not need tobe labeled, but the constants in the equation do need to have the correct units applied.

To find the units it helps to first re-write the equation using just the units, letting some variable such as u (or u1, u2,u3, etc.) stand for the unknown units. The example equation would then become

kg =u1

(s−1)2+ u2

Since any quantities being added or subtracted must have the same units the equation can be split into two equations

kg =u1

(s−1)2and kg = u2

The next step is to solve for u1 and u2, in this case u2 = kg is immediately obvious, u1 will take a bit more effort. Afirst simplification yields

kg =u1

s−2

then multiplying both sides by s−2 gives(s−2)(kg) =

( u1

s−2

)(s−2)

which finally simplifies and solves tou1 = kg · s−2

Inserting the units into the original equation finally yields

m =3.658 kg · s−2

f2− 1.62 kg

which is the final equation with all of the units in place. Checking by choosing a frequency and carrying out thecalculations in the equation will show that it is dimensionally consistent and yields a result in the units for mass (kg) asexpected.


52

Final Exam Description

There will be 100 questions on the final exam. None of them will directly be from chapters 1–11 but topics covered inthose chapters form a foundation for later topics.

# of % ofCategory Topics Chapters Questions Test Total

Circular Motion Gravitation & Satellites 12, 13, 14 9 9% 9%Waves Wave Mechanics 25 11 11% 22%

Sound 26 11 11%Light & Color 27, 28 11 11%

Optics Reflection & Mirrors 29 5 5% 38%Refraction & Lenses 29, 30 13 13%Diffraction & Interference 31 9 9%Electrostatics 32 5 5%

Electricity Electric Fields & Potential 33 5 5% 20%Current 34 5 5%Circuits 35 5 5%

Magnetism Magnetism 36 6 6% 11%Induction 37 5 5%


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Physics Review Notes - Tom · PDF filePhysics Review Notes 2007 ... 3 Chapter 3 — Projectile Motion ... Chapter 4 — Newton